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pre-algebra kk my teach told me to describe and draw things that are about pre-algebra and she told me to write what pre-algebra is... so can anyone help me??
Monday, August 25, 2008 at 9:03pm by damainmind
what does pre-algebra mean?? pre algebra is like a bunch of math that comes before algebra in middle school.
Monday, August 25, 2008 at 9:11pm by GraceHELP! pre-algebra What does this question have to do with pre-algebra? Using an online unscrambling tool, I came up with "proprietors"
Friday, September 5, 2008 at 11:27pm by drwls
pre algebra i told u pre algebra... but i had to give back the text book so i dont have any now
Sunday, June 20, 2010 at 9:38pm by lavena
pre-Algebra Sorry, struggling through pre-Algebra .. :-( .. How do i figure out 5x = -20 ?
Friday, February 19, 2010 at 9:55pm by Emily
Pre-Algebra I do not get this! me and my friend cannot get the "simplified" ecuation part of the homework! Pre-algebra...... tough work
Tuesday, September 15, 2009 at 11:25pm by Maddison
Pre-Algebra the world wobbl e contests? i dunoo pre-algebra sucks
Tuesday, September 15, 2009 at 11:25pm by marinapre-algebra so the 15 - 9=4 okay thanks ms. sue for helping me with my 7th grade pre algebra hw I will probably need more help I will jut post my other questions too.
Sunday, September 16, 2012 at 4:23pm by emily
what does pre-algebra mean?? i have to draw things that have to do with pre algebra but...i dont understand the meaning of it
Monday, August 25, 2008 at 9:11pm by damainmind
algebra 2 thats part of algebra 2 thats easy were doing that now in pre algebra
Wednesday, April 29, 2009 at 1:14pm by Tanisha
pre algebra my grade is 83.17 in pre algebra in college, but the final exam is worth 300 points. If I got a B or B- or a C on the final, what would my grade be overall?
Thursday, March 10, 2011 at 9:20pm by Amber shall
pre-algebra um as in for what is pre-algebra??
Monday, August 25, 2008 at 9:03pm by damainmind
Pre-Algebra its my pre-algebra homework
Tuesday, January 18, 2011 at 9:23pm by TiffanyJ
Pre Algebra random question: i have a pre-algebra unscramble and i need help to unscramble the word the word they gave me was: Seproropit can any of u help me?
Thursday, June 5, 2008 at 5:21pm by coco
Algebra 2 In Kentucky, where i am from, we do algebra and algebra two before pre-cal and calculus. I didnt realize that the problem could be solved in multiple ways and i apologise. But the solution should be done without calculus.
Thursday, February 24, 2011 at 10:33pm by Anon
Pre-Algebra ok well my pre algebra teacher copied some worksheets and its pg 40 and its about some friars in a floral shop buisness. for extra credit we do the joke which is also a way to find out if our answers were correct. there is no 3/8 in my answers and welll yeah thats it.
Tuesday, November 8, 2011 at 10:28pm by Emily
Pre-Algebra Evidently you haven't mastered pre-Algebra since your answer is WRONG. It takes him 4 hours to catch her. While she does ride for 5 hours, it only takes him 4 to catch her.
Wednesday, August 18, 2010 at 6:28pm by Joe Mama
Pre-Algebra: Help I have to do a webquest for pre-algebra and we have to make up a fundraiser thing, I need a list of things that one needs for a charity fundraiser, such as chairs, buffet tables, tables, tents, tablecloths. HELPP
Thursday, March 3, 2011 at 12:20pm by Cherylpre algebra You have to decide you're going to learn how to do algebra.
Wednesday, June 16, 2010 at 10:05pm by Ms. Sue
pre-algebra write an inequality for the sentence. The total t is greater than five. Well your answer is going to be T and the sign for greater than is simply >. So the answer would be T > 5 ok again my name is above. i was just wondering what pre-algebra was? i mean i take the ...
Friday, March 9, 2007 at 8:50am by JoePre-Algebra Sorry, you are PRE-algebra, so you may not know how to solve for x. 8x = 10x - 10 Subtract 10x from both sides. -2x = -10 Divide both sides by -2. x = 5
Wednesday, August 18, 2010 at 6:28pm by PsyDAGtrig is this algebra or pre-algebra
Thursday, February 2, 2012 at 8:14pm by Brianna
Pre Algebra Thank you so much. I have another question :) In my Pre Algebra book, I have a question like this: Draw a line from one vertex to a point on another side to create a triangle. Cut along the line. What do they exactly mean by 'cut along the line'?
Thursday, March 20, 2008 at 6:36pm by Lilypre-Algebra How can you solve algebra equations? The idea is to find a way to isolate the unknown by itself. You can add, subtract, multiply, and divide both </s> sides of the equal signs to do that.
Friday, May 11, 2007 at 5:03pm by Cherlyn
Math oh no Vitaliy, this person is in 7th grade pre-agebra or 8th grade pre-algebra.
Thursday, February 5, 2009 at 6:58pm by haha
PRE-CALCULUS THIS IS FOR ALGEBRA/PRE-CALCULUS MATH. I POSTED IT JUST AS THE TEACHER WROTE IT.
Tuesday, September 13, 2011 at 10:41pm by aLVIN
pre-algebra i don't get them
Thursday, November 14, 2013 at 11:07am by pre-algbra-calc. Answers for Determine The Minimum Distance From The Point (18,20) To The Line 5x+8y=120 Include: Graph, Algebra Of Any Equations, Algebra Of System, Set-up & Steps Of Final Calculations.
Sunday, October 21, 2012 at 11:59pm by OkzyAlgebra II I first want to make it clear that I have only taken Geometry, Algebra I, and pre-Algebra. That seemed to clear things up last time. Question: Matt bought a car at the cost of 25,000 dollars. This car's value goes down 15% every year. 1. What is the decay factor? 2. Write the ...
Thursday, March 17, 2011 at 11:23pm by Anon
pre calculus Pre-Calculus? I wonder what the question is. In a basic algebra course, you would be looking for zeroes, and probably get those by factoring. 2x^2+11x-21=0 (2x-3)(x+7)=0 x= 1.5, or x=-7
Tuesday, December 10, 2013 at 12:39pm by bobpursley
pre algebra what's pre-algebra about this? Sounds like 3rd-grade long division. Take a visit to and you can see all the steps involved with long division
Monday, September 10, 2012 at 11:10pm by Steve
Pre-calc Pre calculus? This is stock Algebra II. put the equations in the form of ax^2+bx+c=0 then factor. for instance, c is already in that form. 2x^2+x-6=0 (2x-3)(x+2)=0 x= 3/2 x=-2 do the others the same method.
Monday, January 25, 2010 at 5:18pm by bobpursley
Math (Algebra/Pre-Algebra) After we took off 20%, doesn't the $15 represent 80% of what it used to cost? so .8x=15 x = 15/.8 = 18.75 test the answer by taking 20% of that, and subtracting it. Do we get $15 ?
Thursday, June 4, 2009 at 8:48pm by Reiny
pre-algebra Use algebra tiles to find the difference. Let one white tile equal +1 and one black tile equal –1.
Thursday, April 5, 2012 at 5:33pm by cakes
pre-algebra Simplify- (-2c+d)(-5)+3(-2c)(-8d) Solve the Equation- y/12= -3 -9+a=11 Here are rules that will help you. I hope it helps. Thanks for asking.
Wednesday, March 7, 2007 at 10:37pm by Janet
pre-algebra Simplify- 2(x+2y)-y 5a+2b+3b-7a 2(2r-2)-8(2r+2) Here are some rules that might be helpful: I hope this helps. Thanks for asking.
Wednesday, March 7, 2007 at 10:30pm by Jackie
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This resource assists the user in reading, constructing, and understanding confidence intervals. Created and published by Gerard E. Dallal, this introductory text aims to get students to read, understand, and write...
This page, by Richard Lowry of Vassar College, will calculate the intercorrelations (r) for any number of variables (V1, V2, V3, etc.) and for any number of observations per variable. Visitors will also find a link to...
Presented by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course "is a study of the basic skills and concepts of elementary algebra, including language and operations covers all of the material outlined by the College Board as necessary to prepare students to pass the...
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The Math Forum is an online community of teachers, students, researchers, parents, educators, and citizens at all levels who have an interest in mathematics and math education. The Math Forum has been consistently recognized as the leader in its field, and continues to provide high quality content and useful features, attracting about 4 million pageviews each month.
The Problems of the Week are designed to challenge students with non-routine problems, and to encourage them to explain their solutions. There are six Problems of the Week (PoWs): Elementary, Middle School, Algebra, Geometry, Trigonometry & Calculus, and Discrete Mathematics. While we will continue to provide Problems of the Week, beginning this fall, a fee will be required to access a "mentored" environment in which every student submission is responded to by a mentor, and students are encouraged to strengthen their solutions.
The Problems of the Week have evolved to include additional useful features including:
a Library of Problems of the Week that organizes the archives for browsing by mathematics and story topic, rates problems for difficulty level, and provides for searching by keyword.
a "Print This Problem" link which allows the problems to be printed with a simple "Math Forum Problem of the Week" header. This feature allows problems to be used without indicating a course or grade level.
teacher accounts which track each student's last posting date as well as correct, bonus, and total submissions. To request an account, follow the "Teacher Account" link for a particular PoW from the Teacher Information page at here is a sample account page;
the Problem of the Week Discussion group at This list has been established to facilitate conversation around the Math Forum's Problems of the Week. We invite input from teachers who have used these problems in their classrooms and from teachers who have questions concerning how they can implement the problems in their curricula.
our first print publication, Problems of the Week, Volume 1, is available exclusively at our conference booth, #2002.
The Math Forum continues to collect, organize, catalog and annotate math-related web sites from diverse sources in the Internet Mathematics Library. You can search or browse through over 7,000 items in the collection, organized under the headings of Mathematics Topics, Resource Types, Mathematics Education Topics or Educational Level. "Drilling down" from a heading takes you to a set of subcategories, selected sites, and all sites in the category.
Ask Dr. Math is an ask-an-expert service in which anyone can pose a math question at any level. A cadre of volunteer 'doctors' select and respond to problems of interest. In addition to a searchable archive of over 5,000 questions and answers, there is:
a set of nearly 50 Frequently Asked Questions, including items about multiplying a negative by a negative, permutations and combinations, the Fibonacci sequence, Pascal's Triangle, and more;
a Classic Problems page, including such favorites as the Tower of Hanoi, or "two trains leave from different cities at the same time ...", or "how large must a group be so the probability of at least two people having the same birthday is ...", etc.;
a Formulas page, which shows formulas for area, perimeter, and volume of a variety of figures, the connections between coordinate systems, trigonometric relationships, and more.
Teacher2Teacher, like a virtual teacher's lounge, is an environment in which questions are asked and opinions are shared about topics across the broad spectrum of interest to teachers, including classroom techniques, activities, resources, etc. The archive contains over 500 questions and their related discussion threads, including public discussions as issues are explored and opinions expressed.
You are encouraged to
join T2T to receive the Teacher2Teacher
Community Update, which contains community news and related items
of interest from the Math Forum. The application form is at
We have over 300,000 pages of content, so this is quite an extensive search field. Given that ours is a full text searcher, you may want to focus a search in a specific area, or use the "that exact phase" and "complete words only" options.
Efficient searching is an art. You will find our Searching Tips and Tricks page helpful, and our Search Features page offers even more detail about such items as the "Starting Points" that are generated for many keywords and topics, and the automatic spell correction. These features are the result of the on-going design efforts to make the search environment more user-friendly. We invite you to contact us to clarify any unresolved confusion or questions.
The Math Forum is committed to building upon the activity of the teachers,
students, and researchers who use it. The Forum provides a platform and the opportunity to share excellent resources and materials with colleagues world wide.
We are particularly pleased to highlight the work of Suzanne Alejandre, including lessons and activities targeted mostly at the middle school level.
Our electronic newsletter is sent out via e-mail once a week to those who subscribe, and is archived on our site. It offers tips about what we have at the Math Forum and how to find it, notes about new items on the site or on the Internet, questions and answers from services like Ask Dr. Math or the Problems of the Week, suggestions for K-12 teachers and students, and pointers to key issues in mathematics and math education.
The Math Forum's discussion archives include many mathematics and math education-related newsgroups, mailing lists, and Web-based discussions, such as the pow-teach discussion mentioned above, as well as math-teach, numeracy, geometry-pre-college, k12.ed.math, sci.math, etc.
There are many ways to contribute to the Math Forum community. Beyond using the various services we provide, many people subscribe to the newsletter, participate in T2T and other discussions, and make suggestions, such as alerting us to other good materials and websites they have discovered. Others find satisfaction in sharing their content as web units or lessons, or showcasing their students' work. Many people volunteer their time and efforts to respond to T2T or Ask Dr. Math questions, while others act as mentors for one of the Problems of the Week.
In what ever ways this might work best for you, please know that you are always welcomed and invited to interact with us in our on-line math ed community center.
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Complete Algebra 1 course online.
Core Algebra 1 synthesizes guided practice worksheets with video instruction and multilevel online followup exams. Today's student is so immersed in
computer and internet technologies that most do not learn efficiently from the old style school methods of reading the textbook and taking notes in class. They
are accustomed to
utilizing the internet as a media for socialization and study which provides instantaneous feedback. We present the core concepts of Algebra 1 in a medium that todays student is familiar with.
This course is designed for:
Current Algebra 1 students needing extra practice or to prestudy
lessons.
Homeschool Algebra 1 students who need guided practice.
Review for Algebra 2 students.
College Algebra 1 students needing review.
Parents who want to review Algebra 1 to assist their children.
All content is provided free of charge.
Core Algebra 1 Content Specifics
12 Units of Study
All Algebra 1 Core Concepts covered
Lessons follow most common current textbooks (PH, McDougal, Glencoe)
39 Sections (Lessons) - Each Lesson Includes
A Video Lesson
Downloadable Notes with the Practice Problems that are covered in video lesson and a solutions page
If you feel the website has helped you and is a valuable internet resource and would like to make a donation towards the maintenance of Core Algebra 1 and towards my plans
for a companion Core Algebra 2 website please click on the donate button and give whatever you feel is appropriate. Thank you!
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Mathematics: About Us
Mathematics is central in the development of the sciences and engineering, statistics, and economics. It is a diverse dynamic field with new important discoveries made constantly.
At Pacific University, mathematics students receive individualized attention from faculty, participate in extracurricular activities sponsored by the department and graduate with a world of opportunity.
Coursework
The mathematics program at Pacific follows the guidelines of the Mathematics Association of America (MAA) and includes courses in all the fundamental areas of undergraduate mathematics including:
Calculus
Linear Algebra
Discrete Mathematics
Mathematical Modeling
Mathematical Probability
Ordinary and Partial Differential Equations
Real Analysis
Complex Analysis
Abstract Algebra
Numerical Analysis
Higher Geometry
All majors, working closely with a faculty member, participate in a senior capstone project to synthesize their mathematics educational experience.
Careers
Mathematics graduates from Pacific University have pursued a wide variety of careers.
Teaching
The mathematics major is especially powerful for students considering middle or high school teaching. Pacific's faculty work directly with the College of Education and with current teachers to ensure that our program is current and appropriate for today's teachers.
Engineering and Technical Work
The mathematics major includes courses in applied mathematics that have helped our graduates enter graduate school in engineering and to obtain good jobs in technical fields.
Finance and Actuarial Science
Many mathematics majors pursue careers in risk assessment, mathematical finance, and actuarial science. These are consistently ranked as the number one careers, offering stability, excitement, and excellent compensation.
Graduate School of Mathematics
Because Pacific's curriculum follows MAA guidelines, students with a mathematics major from Pacific are well-prepared to enter graduate school in mathematics.
Resources
The Mathematics and Computer Science Departments have two computer labs. The math lab has 19 Macintosh computers running OS X and each is outfitted with statistical software and computer algebra systems. For special work, mathematics students can have access to the CS lab. This lab has dual boot Linux/Windows based computers as well as other experimental workstations. Students can also receive accounts on the department's servers. Other departmental resources include:
CBL equipment for real-time data acquisition
Graphing calculators
Web server with space for students pages
Extracurricular Activities
Pacific University and the mathematics faculty sponsor many activities for mathematics students. Some of the current activities are:
Math Club (official ASPU recognized club)
Activities include social events and service opportunities
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Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity
Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity by Joseph C. Kolecki
Publisher: Glenn Research Center 2005 Number of pages: 92
Description: Tensor analysis is useful because of its great generality, computational power, and compact, easy-to-use notation. This monograph is intended to provide a conceptual foundation for students of physics and engineering who wish to pursue tensor analysis as part of their advanced studies in applied mathematics.
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While calculating your GPA for Math majors may not be a complicated task, for others it can be a tedious task, especially if your school uses letters grades with pluses and minuses. Whether your school uses pluses and minuses, the grade scale in the image to the right represents the commonly accepted 4.0 grading scale values for all letter grades.
There are two main types of grading systems, one is weighted the other is non-weighted. Weighted grading systems give each class a defined number of credit hours based on the amount of time the class meets. A class that meets more often gets a higher number of credit hours or 'weight'. Non-weighted grading systems do not assign credit hours to each class, so each class has an equal weight, and credit hours doesn't play a role in calculating your GPA.
As we explained, with schools that use weighted courses, your classes have a defined number of credits assigned to each class. If a certain class meets more than another, they are worth more credits. This means an A in a 4 credit class is worth more than an A in a 3 credit class.
Suppose, for example, you received the following grades for the courses listed below. How do you calculate your GPA?
Note: These steps can be tedious and they're not for everyone. If after reviewing these steps you find them too tedious or want an automated tool, consider downloading a spreadsheet or visiting a website with a GPA calculator, like one we have referenced at the bottom of this article.
Ad
Steps
1
Assign the appropriate scale value for each letter grade. To do this, just match up each letter grade with it's scale value and write it next to the grade. The results of this are shown in the graphic below:
Ad
2
Now that you have the appropriate scale value for each grade, multiply the scale value by the number of credits to get the grade points. The results of this are shown in the graphic below:
3
Add the number of credits together to get the totals credits. In this example, the total credits will be 15.5.
4
Add the grade points together to the total grade points. In this example, the total grade points will be 45.4.
5
To find your GPA, divide your total grade points by the total credits. In this example it would be
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A classic work of American literature that has not stopped changing minds and lives since it burst onto the literary scene, The Things They Carried is a ground-breaking meditation on war, memory, imagination, and the redemptive power of storytelling. The Things They Carried depicts the men of Alpha Company: Jimmy Cross, Henry Dobbins, Rat Kiley,...Tough Test Questions? Missed Lectures? Not Enough Time?
Fortunately, there's Schaum's. This all-in-one-package includes more than 550 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 30 detailed videos featuring Math instructors who explain how to solve the most commonly... more...
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Mathematics
APBS is a software package for the numerical solution of the Poisson-Boltzmann equation, a popular continuum model for describing electrostatic interactions between molecular solutes over a wide range of length scales. <
There are countless programs out there that solve complex calculus related problems or simple algebraic equations.My plans for this project is to make a complete Algebra and Calculus suite that would cater to students of all levels.
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Mathematics and Its History
9780387953366
ISBN:
0387953361
Edition: 2 Pub Date: 2001 Publisher: Springer Verlag
Summary: From the reviews of the first edition: "[This book] can be described as a collection of critical historical essays dealing with a large variety of mathematical disciplines and issues, and intended for a broad audienceA? we know of no book on mathematics and its history that covers half as much nonstandard material. Even when dealing with standard material, Stillwell manages to dramatize it and to make it worth rethin...king. In short, his book is a splendid addition to the genre of works that build royal roads to mathematical culture for the many." (Mathematical Intelligencer) This second edition includes new chapters on Chinese and Indian number theory, on hypercomplex numbers, and on algebraic number theory. Many more exercises have been added, as well as commentary to the exercises explaining how they relate to the preceding section, and how they foreshadow later topics.
Stillwell, John is the author of Mathematics and Its History, published 2001 under ISBN 9780387953366 and 0387953361. One hundred sixty seven Mathematics and Its History textbooks are available for sale on ValoreBooks.com, fifteen used from the cheapest price of $24.45, or buy new starting at $33.78.[read more]
Ships From:Salem, ORShipping:Standard, ExpeditedComments:Has minor wear and/or markings. SKU:9780387953366-3-0-3 Orders ship the same or next business day... [more]
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Middle School Mathematics: Course 2
Teacher: Mr. Vanden Bogerd
Phone: (269) 267-5923
Email: gvandenbogerd@zionchristian.net
Website:
COURSE DESCRIPTION: This course is designed to prepare middle school students for success in both
Algebra and Geometry. The major topics covered within this course include: fractions, decimals, percents, ratios,
proportions, statistics, probability, integers, algebra, geometry, measurement, linear equations and functions. During
this course students will be engaged in problem solving, justifying algorithms, analyzing data, explaining
mathematical relationships, and making applications to real-world situations.
REQUIRED TEXT: Mathematics: Applications and Concepts, Course 2
COURSE OBJECTIVES: At the end of this course students should be able to:
1. Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
2. Understand the meaning of operations and how they relate to each other.
3. Compute fluently and make reasonable estimates.
4. Understand algebraic patterns, relations, and functions.
5. Represent and analyze mathematical situations and structures using algebraic symbols.
6. Use mathematical models to represent and understand quantitative relationships.
7. Analyze change in various contexts.
8. Analyze characteristics and properties of two- and thee-dimensional geometric shapes and develop
mathematical arguments about geometric relationships.
9. Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
10. Apply transformations and use symmetry to analyze mathematical situations.
11. Use visualization, spatial reasoning, and geometric modeling to solve problems.
12. Understand measurable attributes of objects and the units, systems, and processes of measurement.
13. Apply appropriate techniques, tools and formulas to determine measurements.
14. Formulate questions that can be addressed with data and collect, organize, and display relevant data to
answer them.
15. Select and use appropriate statistical models to analyze data.
16. Develop and evaluate inferences and predictions that are based on data.
17. Understand and apply basic concepts of probability.
*Course objectives are based on the NCTM Principles and Standards for School Mathematics
COURSE LAYOUT: This course is divided into five units which are subdivided into a total of twelve
chapters. Three chapters will be covered each quarter of the school year.
Unit 1 Decimals, Algebra, and Statistics
Chapter 1 Decimal Patterns and Algebra
Chapter 2 Statistics: Analyzing Data
Unit 2 Integers and Algebra
Chapter 3 Algebra: Integers
Chapter 4 Algebra: Linear Equations and Functions
Unit 3 Fractions
Chapter 5 Fractions, Decimals, and Percents
Chapter 6 Applying Fractions
Unit 4 Proportional Reasoning
Chapter 7 Ratios and Proportions
Chapter 8 Applying Percent
Chapter 9 Probability
Unit 5 Geometry and Measurement
Chapter 10 Geometry
Chapter 11 Geometry: Measuring Two-Dimensional Figures
Chapter 12 Geometry: Measuring Three-Dimensional Figures
1
EVALUATION AND GRADING OF ASSIGNMENTS PER QUARTER:
Approximate Points per Category Description
Tests: 300 points There will be an average of three tests per quarter. One
[about 40% of final grade] after each chapter.
(100 points per test)
Homework: 400 points There will be an average of five graded homework
[about 50% of final grade] assignments per week. These will be graded for accuracy
(10 points per assignment) and/or completeness. Online assignments will count towards
your homework grade.
Start Assignments: 80-90 points The "start assignments" packet will be collected at the
[about 10% of final grade] end of the quarter. Points will be deducted for wrong
(2 points per problem) answers to any problem that we went over in class.
Assignment Books/Journals Assignment books and/or math journals may be graded
[grade will be factored into randomly to verify that they are being used.
the homework grade]
Extra Credit: (30 points) There will be several opportunities for earning extra
(10 points per assignment) credit. The most important of these is the standardized
tests that may be completed at the end of each chapter.
These are a great way to earn extra points and to prepare
for the semester exams.
GRADING SCALE:
PERCENT LETTER G.P.A.
96-100 A 4.00
93-95 A- 3.67
90-92 B+ 3.33
87-89 B 3.00
84-86 B- 2.67
81-83 C+ 2.33
78-80 C 2.00
75-77 C- 1.67
72-74 D+ 1.33
69-71 D 1.00
66-68 D- 0.67
0-65 F 0.00
Note that the grade for each of your cumulative semester exams will be factored into the final grade for the
semesters. These exams will be worth 20% of each of your semester grades. The final exam for this course will
include material from both semesters.
ON-LINE MATH RESOURCES: There will be occasional online assignments including chapter readiness
assignments and chapter practice tests. These may be accessed by visiting my website at
You will find the assignments on the Middle School Course 2 math page. Each assignment must be completed within 24
hours from when it is assigned. An online assignment may be done over multiple times before you email your score to me
at gvandenbogerd@zionchristian.net. Extensions of due dates will be granted for online assignments in the event that you
encounter technical difficulties that are beyond your control. Students without internet access are welcome to request a
printed copy of the online assignments.
2
GENERAL COURSE GUIDELINES: The following guidelines are stated to the end that all things might
be done decently and in order within the middle school mathematics (Course 2) classroom. These guidelines do not
intend to detract from or to replace any of the school's policies as they are stated within the Family Handbook.
1. Assignments specified as homework are to be completed by the beginning of the class period for which
they are due. Assignments that are turned in after the beginning of the class period will be deducted 10%
for being one day late. At the discretion of the instructor, and/or if students have no legitimate excuse for
submitting the assignment late, assignments will be graded as zeros if they are not turned in by the
beginning of the next class period. Contact with the student's parents will generally result after a student
receives a zero for a missing assignment.
2. The student is expected to complete written homework without help from others or the use of a calculator,
unless this is specified by the teacher. Calculators will be necessary for most second semester material.
Using a calculator on first semester material without permission will be treated as a violation of academic
integrity.
3. All math assignments are to be completed in pencil. Infringement of this guideline may result in the student
receiving a tardy on account of being inadequately prepared for class.
4. Unless specified, answers to math problems must include the student's work. Failure to show work may result
in the deduction of points for the specific assignment. Please do not erase your work.
5. The student will be required to correct and return any homework assignment that reveals a specific pattern
of error, or when the assignment receives a score below 70% accuracy. Credit will be given for each
corrected answer.
6. When students are permitted or required to work in groups, each student is responsible for submitting
his/her own work.
7. Appropriate cooperation and diligence is mandatory for all group work settings.
8. Students are expected to use a math journal on a daily basis for recording strategies and methods for problem
solving as well as examples of problems provided by the teacher.
9. Students are to come prepared to class with all of the following items:
a. Mathematics: Applications and Concepts, Course 2 textbook
b. Math Journal (a spiral notebook designated for taking notes in math)
c. An inexpensive scientific calculator (Required every day for 2nd Semester)
d. Several previously sharpened pencils and an eraser
e. Assignment book
f. Any notes or assignments that are necessary for participation in class
Failure to bring to class the above mentioned items will result in a tardy.
FASTT MATH: Students are required to complete at least one session of FASTT MATH each week.*
Students must request permission before using classroom computers.
Classroom computers may only be used during the individual work time. (All group work must be
completed before students may do FASTT MATH.)
Unless permitted, classroom computers may only be used for FASTT MATH during the math class. Improper
use of classroom computers during math class may result in a grade reduction in the FASTT MATH category.
* This requirement is contingent upon the availability of computers and the functionality of the software.
QUESTIONS: If you have any questions, comments, suggestions, and/or concerns; if you would like additional
help; or if you would like to discuss anything with me, please come and let me know during the noon hour break or
right before or after class. You are also welcome to contact me at any time by phone at 269-267-5923 or by email at
gvandenbogerd@zionchristian.net
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More About
This Textbook
Overview
This first-year text offers a straightforward introduction to integral and differential calculus. Provides clear explanations of the main concepts of the calculus, including a brief review of algebra. Also contains excellent problem sets. Offers careful, well-organized development of limit, first derivative and the definite and indefinite integrals, supported by numerous graphs, diagrams and applications-oriented examples and problems. Also contains sections on differential equations and numerical
| 677.169 | 1 |
""This book would be a great tool for helping [today's future elementary teachers] acquire a 'gut level' understanding of mathematics concepts."" - ...Show synopsis""This book would be a great tool for helping [today's future elementary teachers] acquire a 'gut level' understanding of mathematics concepts."" - Hester Lewellen, Baldwin-Wallace College, OH ""The writing in this text is very clear and would easily be understood by the intended audience. The real-world examples put the various math concepts into a context that is easily understood. The vignettes at the beginning of each chapter are interesting and they get the reader to begin thinking about the math concepts that will follow. Each of the chapters seem to build on one another and the author often refers back to activities and concepts from previous chapters which is meaningful to the reader because it lets the reader know that the information they are learning builds their conceptual understanding of other mathematical concepts. "" - Melany L. Rish, University of South Carolina, Aiken Organized around five key concepts or "powerful ideas" in mathematics, this book presents elementary mathematics content in a concise and nonthreatening manner for teachers. Designed to sharpen teachers' mathematics pedagogical content knowledge, the friendly writing style and vignettes relate math concepts to "real life" situations so that they may better present the content to their students. The five "powerful ideas" (composition, decomposition, relationships, representation, and context) provide an organizing framework and highlight the interconnections between mathematics topics. In addition, the book thoroughly integrates discussion of the five NCTM process strands. Features: Icons highlighting the NCTM process standards appear throughout the book to indicate where the text relates to each of these. Practice exercises and activities and their explanations reinforce math concepts presented in the book and provide an opportunity for reflection and practice. Concise, conversational chapters and opening vignettes present math contents simply enough for even the most math-anxious pre-service teachers205493753-5-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions. ISBN: 9780205493753.
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Mathematics: Calculus, Statistics, and Logic Courses
Approved courses introduce students to or extend their knowledge of precalculus, calculus, discrete mathematics, probability, statistics and/or data analysis. Courses may be offered in the Department of Mathematics and Statistics and in other departments that have expertise in quantitative reasoning and data analysis and that offer appropriate courses, particularly in statistics or discrete structures.
A student who has achieved a score of 85 or above on the Regents "Math B" Exam (former "Mathematics Course III" Exam) or on a recognized standardized examination indicating readiness to enter precalculus will be considered to have fulfilled this requirement.
Learning Objectives for Mathematics: Calculus, Statistics, and Logic
Courses in Calculus enable students to demonstrate:
an understanding of basic mathematical functions and their graphical representations together with an ability to understand how many quantities of interest in mathematics, the sciences, and the social sciences can be modeled by functions and their properties understood graphically;
an ability to calculate derivatives and use them to analyze graphs, solve problems (growth/decay, optimization, rates of change), and make approximations;
an ability to use integrals to calculate quantities of interest (area, volume, work, moments, probabilities).
an ability to extract information from graphs and other displays (such as scatter plots and histograms);
an ability to choose and appropriate statistical procedure for evaluation of various types of data;
an ability to calculate confidence intervals (mainly for means of one and two sample tests) and to set up and interpret the result of standard hypothesis tests, which require the use of tables (mainly of the normal and t distributions).
Courses in Logic enable students to demonstrate:
an ability to translate sentences and arguments from English into the formal systems and to demonstrate or prove arguments within the formal systems;
an ability to evaluate formal properties of sentences (and sets of sentences) using truth tables and to prove formalized arguments;
an ability to examine, interpret, and represent the logical structure of the original English expressions and draw communicable conclusions (e.g. that an argument is invalid).
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Math Survival Guide Tips and Tricks for Science Students
9780471270546
ISBN:
0471270547
Edition: 2 Pub Date: 2003 Publisher: Wiley & Sons, Incorporated, John
Summary: This second edition of 'Math Survival Guide' provides tips for science students in the form of a quick reference/update guide. It uses an approachable tone and appropriate level and includes good problem sets.
Appling, Jeffrey R. is the author of Math Survival Guide Tips and Tricks for Science Students, published 2003 under ISBN 9780471270546 and 0471270547. Five hundred sixty two Math Survival Guide Tips an...d Tricks for Science Students textbooks are available for sale on ValoreBooks.com, one hundred fifty used from the cheapest price of $8.99, or buy new starting at $32.36 [more1270547 Your purchase benefits those with developmental disabilities to live a better quality of life. Your purchase benefits those with developmental disabilities to live [more]
0471270547 Your purchase benefits those with developmental disabilities to live a better quality of life. Your purchase benefits those with developmental disabilities to live a better quality of life specifically designed as a study guide and resource for science students confronted with mathematics that they need extra help on. This math skills review and pr [more]
This book is specifically designed as a study guide and resource for science students confronted with mathematics that they need extra help on. This math skills review and practice guide is written in a clear, accessible manner to bring readers up to
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ISBN13:978-0534432232 ISBN10: 0534432239 This edition has also been released as: ISBN13: 978-0534381257 ISBN10: 0534381251
Summary: Helping students grasp the "why" of algebra through patient explanations, Hirsch and Goodman gradually build 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 definit...show moreions 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. ...show less
Basic Definitions: The Real Numbers and the Real Number Line Operations with Real Numbers Algebraic Expressions Translating Phrases and Sentences into Algebraic Form First-Degree Equations and Inequalities Chapter Summary, Review Exercises, and Practice Test
Straight Lines and Slope Equations of a Line and Linear Functions as Mathematical Models Linear Systems in Two Variables Graphing Linear Inequalities in Two Variables Chapter Summary, Review Exercises, and Practice Test
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1
Mathematically Correct:
Finding the Best Equation for U.S. Math Instruction
Katherine Vazquez
Brooklyn College
Education 7201-Fall 2011
2
Table of Contents
Abstract
Introduction……………………………………………………………………………3
Statement of the Problem……………………………………………………….3
Review of the Related Literature………………………………………………..3-6
Statement of the Hypothesis…………………………………………………….6
Methods
Participants
Instruments
Experimental Design
Procedure
Results
Discussion
Implications
References……………………………………………………………………………...7-9
Appendices……………………………………………………………………………10-12
3
Introduction
Math and technology education are becoming increasingly more important in today's
advanced, gadget-driven society. The highest paid jobs are almost unequivocally science based,
with engineering and medicine consistently ranking at the top of the charts. Additionally,
globalization has made competition for such vocations of high occupational prestige exceedingly
stiff. In order to ensure that future generations of American children are productive and viable
citizens in the global economy, it is imperative that they are immersed in sound math instruction
beginning at the elementary school years. Only once they gain mastery of basic computational
skills can they have a chance at excelling at the more abstract levels of problem solving required
at the high school and college level and beyond.
Statement of the Problem
International mathematics assessments indicate that United States students consistently
rank far behind their peers in similarly developed countries. Scores on the National Assessment
of Educational Progress, or NAEP, demonstrate that far too few U.S. students are at or above the
proficient level in math and science (Epstein & Miller, 2011). New techniques that flout tried
and true math teaching methods are a key source of the disparity. Education reformers,
representing the education establishment, believe the learning "process" is more important than
memorizing core knowledge. They see self-discovery as more important than getting the right
answer. Traditionalists, consisting mainly of parent groups and mathematicians, advocate
teaching the traditional algorithms. They advocate clear, concrete standards based on actually
solving math problems. The destination - getting the right answer - is important to traditionalists.
The textbook that has become the gold standard for reformers is called Everyday Math. It
is deeply flawed in its approach. It does not teach addition with regrouping and instead uses
cumbersome, time-consuming, less efficient, more laborious, non-standard "partial sums"
method. It also discourages the practice of standard algorithms for multiplication and division.
Here too it incorporates cumbersome, time-consuming, less efficient, more laborious, unduly
complicated "extended facts," "partial products," and "lattice" methods. A formal introduction to
division algorithms is not included and crutches (e.g., counters, arrays, drawings) in division are
never dropped.
Reform/Constructivist Math Curricula
Overview of Reform Math Literature
Herrera & Owens (2001), note that the most recent movement to revolutionize math
teaching in the United States is NCTM Standards-based reform. The reform Standards do not
list specific topics to be covered by the end of each grade. Instead, guidelines are provided with
examples intended to present a unique conception of math content. One of the benefits of the
movement is the push to make concrete connections between mathematics and the real world
paramount (Varol & Farran, 2007). There is also more of an emphasis on higher order
processing through problem solving, communication, and reasoning. The shift from direct,
algorithm-based instruction to Standards-based reform is underpinned by a new emphasis on
constructivism and conceptual knowledge over procedural knowledge. In the past, the primary
mathematics computation in early school years was based on the pen and paper algorithm (Varol
& Farran, 2007). However, modern reformers now realize the importance of mental
computation.
Reform mathematics is also known as research-based mathematics because its policies
are largely aimed at ensuring that efforts to reform math education are rooted in current and
high-quality scientific knowledge about what content students should learn, how they should
4
learn such content, and how they should be assessed (Superfine, Kelso, & Beal, 2010). Whereas
many see the reforms as merely a fad, advocates of the movement look for data to back up
proposed changes. Fortunately, these changes have been around long enough to be empirically
evaluated (see "Field Testing" below).
Reform Instructional Methods
Fraivillig, Murphy, & Fuson, (1999) conducted a case study of first grade teacher, Ms.
Smith's, use of the reform text, Everyday Math. Her successful strategies included eliciting
students' solution strategies, facilitating their responses, supporting conceptual understanding,
and extending mathematical thinking. She encouraged students not to worry about answers per
se, but instead to collaborate and explore various problem-solving tactics. This behavior was
consistent with the Moyer, Cai, Wang, & Nie, (2011) study that found about twice as many
reform lessons as traditional lessons are structured to use group work as a method of instruction.
This is advantageous to students because teachers whose goal it is to foster their students'
interests are more likely to use cooperative activities in math (Durik & Eccles, 2006).
Ma & Singer-Gabella (2011) analyzed routines in reform classes. According to them, a
typical teacher script in a reform classroom might be as follows:
I would like for you to solve this problem in as many ways as you can come up
with. I will give you a few minutes to think about it. You can talk to other people if
you like and then we'll look at some of the methods by which you've solved the problem.
A book has 64 pages; you've read 37 of those pages, how many pages do you have left to
read? Be sure that for any method you use that you can explain how you did it in terms of
quantity of pages. Come up with as many ways of solving it as you can. (p. 13)
Reform Theorists/Practitioners
The standards are based upon the learning theory of Constructivism (Chung, 2004).
Constructivism is supported by cognitive theorists, such as Jean Piaget, Jerome Bruner, Zoltan
Dienes, and Lev Vygotsky. Notably, Jean Piaget's intellectual development (sensorimotor,
preoperational, concrete operational, and formal operational) and Jerome Bruner's learning
modes (enactive, iconic, and symbolic) provide demonstrations of constructivism in school-age
children. Constructivist ideology focuses on processes and the use of manipulatives. Students
should be introduced to new concepts in three ways to accomplish representation: action
(enactive), visual pictures (iconic), and through the use of words (symbolic). This is meant to
help students transition from concrete to abstract levels of understanding.
Field Testing Reform Math: What the Research Shows at the Elementary Level
Carrol (1997), found that third grade students across 26 reform curriculum classrooms (as
per use of the Everyday Math textbook) scored well above (64 points greater) the state median
score on an Illinois State Mathematics Assessment. Moreover, 14 of achievement in the classes
containing students who had been immersed in the Everyday Math curriculum since kindergarten
was even higher, 75 points above the state score. This suggests a positive longitudinal effect of
the curriculum. This is in accordance with other research (Mong & Mong, 2010) indicating that
the social validity of an intervention may be affected by the time involved.
A flaw in Carrol's study is that the author does not indicate what the SES of students in
the "traditional classrooms" was in comparison to those in the reform classes. This might indeed
be a confounding variable, because students in the traditional classrooms were all from Chicago-
a place known to be plagued by poverty and high dropout rates.
Fuson, Carroll, & Drueck (2000), determined that Everyday Math third graders outscored
traditional U.S. students on place value and numeration, reasoning, geometry, data, and number-
5
story items. The study is not completely reliable, however. Researchers were not able to match
Everyday Math curriculum schools with comparable ones, and therefore chose to use data from
existing studies to provide comparisons. Obviously, this is a weaker comparison than using fresh
scores and evaluations.
Crawford and Snider (2000), conducted a two-year study conducted in two fourth grade
classrooms investigated the effectiveness of two mathematics curricula. Results found that a
reform program based on the text Connecting Math Concepts, resulted in significantly higher
student scores on mathematics tests than the use of a traditional math basal textbook. While in
this instance, the reform text used did yield higher scores, it is important to note that the specific
book in question is not nearly as widespread across elementary schools as its reform counterpart,
the ubiquitous text Everyday Math.
Field Testing Reform Curricula in Middle School and Beyond
There are a couple of studies that suggest reform math might be best implemented in the
middle school grades and beyond, when math becomes more abstract and conceptually oriented.
Cai, Wang, Moyer, Wang, & Nie (2011) determined that for algebra, the use of reform
curriculum contributed significantly to problem-solving growth and students' ability to represent
problem situations. Similarly, in Texas, Vega (2011) found 9th grade ELLs, 9th grade
economically disadvantaged students, and 11th grade African American students who were
reform taught from 2003-2004 were significantly outperformed those traditionally taught.
Traditional/Procedural Math Curricula
Overview of Traditional Math Curricula
Traditionalists eschew the reform notion that students can not only construct their own
understandings of mathematics, but also actually reinvent significant mathematics if given a
chance (Frykholm, 2004). Cognitive ability as well as math fluency play an important role in
mathematical skills. Understanding the relationship between cognitive abilities and mathematical
skills is imperative to teaching effective arithmetic skills (Ramos-Christian & Schleser, 2008).
Traditionalists adhere to the belief that domain-specific mathematical problem-solving
skills can be taught by emphasizing worked examples of problem-solution strategies. A worked
example provides problem-solving steps and a solution for students. Direct, explicit instruction is
vital in all curriculum areas, especially areas that many students find difficult and that are critical
to modern societies. Mathematics is such a discipline. Minimal instructional guidance in
mathematics leads to minimal learning. In short, traditionalists rely on research indicating that
they can teach aspiring mathematicians to be effective problem solvers only by helping them
memorize a large store of domain-specific schemas (Sweller, Clark, & Kirschner, 2010).
Traditional Instructional Methods
In a traditional framework, mathematical problem-solving skill is acquired through a
large number of specific mathematical problem-solving strategies relevant to particular
problems. Studying worked examples interleaved with practice solving the type of problem
described in the example reduces unnecessary working-memory load that prevents the transfer of
knowledge to long-term memory. The improvement in subsequent problem-solving performance
after studying worked examples rather than solving problems is known as the worked-example
effect (Sweller, Clark, & Kirschner, 2010). The didactic teaching world is highly ritualized and
features procedures presented by teachers, with students practicing those procedures alone. For
this reason, Son & Senk (2010), report multistep computational problems to be more common in
traditional textbooks than in reform ones. Traditional textbooks also excel over reform
6
pedagogies in providing more opportunities to practice number sense skills (Sood & Jitendra,
2007).
Traditional Theorists/Practitioners
Sandra Stotsky, Professor of Education Reform at the University of Arkansas, is a
staunch traditionalist. She, educated parents, and prominent mathematicians voice objections to
the stress on calculator use in the early grades, the over-emphasis on student-developed
algorithms at the expense of standard algorithms, and the de-emphasis at the high school level on
computation in algebra and proof in Euclidean geometry (Stotsky, 2007). Countries like
Singapore and Korea, which consistently outperform American students, also are proponents of
traditional, rigorous curricula that focus on procedural knowledge and sound, well-known
algorithms.
Field Testing Traditional Math: What the Research Shows at the Elementary Level
Three Research studies strongly indicate the efficacy of employing traditional texts.
Hook, Bishp, & Hook (2007) established that students in California were shown to make
statistically significant gains in math performance over five years of utilizing a text based on the
six leading TIMMS math countries in Asia and Europe (which are highly traditionalist oriented).
Agodini and Harris (2010) found that across 39 schools first graders using the traditional text,
Saxon Math, performed 0.30 SD higher than reform "Investigations" students and 0.24 SD
higher than "SFAW" students. Finally, Poncy, McCallum, and Schmitt (2010), utilized an
alternating treatments design to compare a traditionalist behavioral intervention, "Cover, Copy,
and Compare" (CCC), to an intervention from a reform-oriented resource, "Facts That Last"
(FTL). Results demonstrated that CCC led to increases in math-fact fluency, whereas the class-
wide response to FTL activities did not differ from the control condition. Two months post-
intervention, maintenance data revealed that the fluency increases associated with CCC were
sustained.
Field Testing Traditional Curricula Abroad
In the Netherlands, Kroesbergen, Van Luit, and Maas (2004) compared the effects of
smallgroup constructivist and explicit mathematics instruction in basic multiplication on low-
achieving students' performance and motivation. A total of 265 students (aged 8-11 years) from
13 general and 11 special elementary schools for students with learning and/or behavior
disorders participated in the study. The experimental groups received 30 minutes of reform or
traditional instruction in groups of 5 students twice weekly for 5 months. Pre- and posttests were
conducted to compare the effects on students' automaticity, problem-solving, strategy use, and
motivation to the performance of a control group who followed the regular curriculum. Results
showed that the math performance of students in the traditional instruction condition improved
significantly more than that of students in the constructivist condition
Research Hypotheses
HR1: 28 4th grade students at O'Neill Elementary School in Central Islip, NY who are
immersed in traditional algorithms are expected to yield higher scores on a mathematical
assessment gauging two digit multiplication skills than those who are exposed to reform math
pedagogies (Everyday Math).
HR2: 28 4th grade students at O'Neill Elementary School in Central Islip, NY who are
taught traditional algorithms will achieve higher scores on a mathematical assessment gauging
subtraction with regrouping skills than those who are taught primarily through reform texts
(Everyday Math).
7
References
Agodini, R, & Harris, B. (2010). An experimental evaluation of four elementary school math
curricula. Journal of Research on Educational Effectiveness, 3, 199-253.
Cai, J, Wang, N, Moyer, J., Wang, C., & Nie, B. (2011). Longitudinal investigation of the
curricular effect: An analysis of student learning outcomes from the LieCal Project in the
United States. International Journal of Educational Research, 50, 117-136.
Carroll, W. M. (1997). Results of third-grade students in a reform curriculum on the Illinois state
mathematics test. Journal for Research in Mathematics Education, 28, 237-242.
Chung, I. (2004). A comparative assessment of constructivist and traditionalist approaches to
establishing mathematical connections in learning multiplication. Education, 125, 271-
278.
Crawford, D. & Snider, V. (2000). Effective mathematics instruction: The importance of
curriculum. Education and Treatment of Children, 23, 122-142.
Durik, A. & Eccles, J. (2006). Classroom activities in math and reading in early, middle, and
late elementary school. Journal of Classroom Interaction, 41, 33-41.
Epstein, D. & Miller, R. (2011). Slow off the mark: Elementary school teachers and the crisis in
STEM education. Education Digest: Essential Readings Condensed for Quick Review,
77, 4-10.
Fraivillig, J., Murphy, L., & Fuson, K. (1999). Advancing children's mathematical thinking in
everyday mathematics classrooms. Journal for Research in Mathematics Education, 30
148-170.
Frykholm, J. (2004).Teachers' tolerance for discomfort: Implications for curricular reform in
mathematics. Journal of Curriculum and Supervision, 19, 125-149.
Fuson, K., Carroll, W., & Drueck, J. (2000). Achievement results for second and third graders
using the standards-based curriculum everyday mathematics. Journal for Research in
Mathematics Education, 31, 277-295.
Herrera, T. & Owens, D. (2001). The "new new math"?: Two reform movements in mathematics
education. Theory into Practice, 40, 84-92.
Hook, W., Bishop, W., & Hook, J. (2007). A quality math curriculum in support of effective
teaching for elementary schools. Educational Studies in Mathematics, 65, 125-148.
Kroesbergen, E. H.,Van Luit, J. E. H., & Maas, C. J. M. (2004). Effectiveness of explicit and
8
constructivist mathematics instruction for low-achieving students in the Netherlands.
Elementary School Journal, 104, 233-253.
Ma, J. & Singer-Gabella, M. (2011). Learning to teach in the figured world of reform
mathematics: Negotiating new models of identity. Journal of Teacher Education 62, 8-
22.
Mong, M. & Mong, K. (2010). Efficacy of two mathematics interventions for enhancing fluency
with elementary students. Journal of Behavioral Education, 19, 273-288.
Moyer, J. C., Cai, J., Wang, N., & Nie, I. (2011). Impact of curriculum reform: Evidence of
change in classroom practice in the United States. International Journal of Educational
Research, 50, 87-99.
Poncy, B. C., McCallum, E., & Schmitt, A. J. (2010). A comparison of behavioral and
constructivist Interventions for increasing math-fact fluency in a second-grade classroom.
Psychology in the Schools, 47, 917-930.
Ramos-Christian, V., Schleser, R., & Varn, M. (2008). Math fluency: Accuracy versus speed in
preoperational and concrete operational first and second grade children. Early Childhood
Education Journal, 35, 543-549.
Son, J. & Senk, S. (2010). How reform curricula in the USA and Korea present multiplication
and division of fractions. Educational Studies in Mathematics, 74, 117-142.
Sood, S. & Jitendra, A. (2007). A comparative analysis of number sense instruction in reform-
based and traditional mathematics textbooks. Journal of Special Education, 4, 145-157.
Superfine, A. C., Kelso, C., & Beal, S. (2010). Examining the process of developing a research-
based mathematics curriculum and its policy implications. Educational Policy, 24, 908-
934.
Stotsky, S. (2007). The Massachusetts math wars. Prospects: Quarterly Review of Comparative
Eduation, 37, 489-500.
Sweller, J., Clark, R., & Kirschner, P. (2010). Mathematical ability relies on knowledge, too.
American Educator, 34, 34-35.
Varol, F. & Farran, D. (2007). Elementary school students' mental computation proficiencies.
Early Childhood Education Journal, 35, 89-94.
Vega, T. & Travis, B. (2011). An investigation of the effectiveness of reform mathematics
9
curricula analyzed by ethnicity, socio-economic status, and limited English proficiency.
Mathematics and Computer Education, 45, 10-16.
10
Appendix A: Parent Consent form
December 4, 2011
Dear Parent/Guardian,
I am currently a graduate student at Brooklyn College. This semester I am in the process of
completing an action research project as one of the requirements for a Research I course. I would
like to invite your child to participate in a Comparative Research Study that will be conducted
during the school year. Therefore, I am requesting your permission to gather data and incorporate
the information in my Master's Thesis. If you decide to allow your child to participate, he/she
may be required to complete questionnaires, demographic surveys, achievement measurements
and participate in possible observations. Through this study, I hope to learn about the impact of
different math curricula on student performance.
Any information that is obtained in connection with this study and that can be identified with
your child will remain confidential and will not be disclosed. The participants will be kept
confidential by assuring that all names remain anonymous.
If you have any questions or concerns, please feel free to contact me via email at
kvaz610@gmail.com. Thank you in advance for your cooperation and support.
Sincerely,
Katherine Vazquez
I ___________________________ have read and understand the information provided above. I
Parent/Legal Guardian Signature
willingly agree to allow my child to participate in this research project.
11
Appendix B: Principal Consent Form
December 4, 2011
Dear PrincipalPrincipal Signature
willingly agree to allow my school to participate in this research project.
12
Appendix C: Teacher Consent Form
December 4, 2011
Dear TeachersTeacher Signature
willingly agree to allow my students to participate in this research
| 677.169 | 1 |
0521709830
9780521709835
Algebraic and Analytic Geometry:This textbook, for an undergraduate course in modern algebraic geometry, recognizes that the typical undergraduate curriculum contains a great deal of analysis and, by contrast, little algebra. Because of this imbalance, it seems most natural to present algebraic geometry by highlighting the way it connects algebra and analysis; the average student will probably be more familiar and more comfortable with the analytic component. The book therefore focuses on Serre's GAGA theorem, which perhaps best encapsulates the link between algebra and analysis. GAGA provides the unifying theme of the book: we develop enough of the modern machinery of algebraic geometry to be able to give an essentially complete proof, at a level accessible to undergraduates throughout. The book is based on a course which the author has taught, twice, at the Australian National University.
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Rent Algebraic and Analytic Geometry 1st edition today, or search our site for Amnon textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cambridge University Press.
| 677.169 | 1 |
GAUSS in the Classroom
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| 677.169 | 1 |
1
00:00:00 --> 00:00:06
2
3
4
00:00:07 --> 00:00:13
OK, let's get started.
I'm assuming that,
5
00:00:10 --> 00:00:16
A, you went recitation
yesterday, B,
6
00:00:13 --> 00:00:19
that even if you didn't,
you know how to separate
7
00:00:17 --> 00:00:23
variables, and you know how to
construct simple models,
8
00:00:21 --> 00:00:27
solve physical problems with
differential equations,
9
00:00:25 --> 00:00:31
and possibly even solve them.
So, you should have learned
10
00:00:31 --> 00:00:37
that either in high school,
or 18.01 here,
11
00:00:35 --> 00:00:41
or, yeah.
So, I'm going to start from
12
00:00:38 --> 00:00:44
that point, assume you know
that.
13
00:00:42 --> 00:00:48
I'm not going to tell you what
differential equations are,
14
00:00:47 --> 00:00:53
or what modeling is.
If you still are uncertain
15
00:00:51 --> 00:00:57
about those things,
the book has a very long and
16
00:00:56 --> 00:01:02
good explanation of it.
Just read that stuff.
17
00:01:00 --> 00:01:06
So, we are talking about first
order ODEs.
18
00:01:06 --> 00:01:12
ODE: I'll only use two
acronyms.
19
00:01:08 --> 00:01:14
ODE is ordinary differential
equations.
20
00:01:12 --> 00:01:18
I think all of MIT knows that,
whether they've been taking the
21
00:01:17 --> 00:01:23
course or not.
So, we are talking about
22
00:01:21 --> 00:01:27
first-order ODEs,
which in standard form,
23
00:01:25 --> 00:01:31
are written,
you isolate the derivative of y
24
00:01:29 --> 00:01:35
with respect to,
x, let's say,
25
00:01:31 --> 00:01:37
on the left-hand side,
and on the right-hand side you
26
00:01:36 --> 00:01:42
write everything else.
You can't always do this very
27
00:01:42 --> 00:01:48
well, but for today,
I'm going to assume that it has
28
00:01:47 --> 00:01:53
been done and it's doable.
So, for example,
29
00:01:50 --> 00:01:56
some of the ones that will be
considered either today or in
30
00:01:56 --> 00:02:02
the problem set are things like
y prime equals x over y.
31
00:02:01 --> 00:02:07
That's pretty simple.
32
00:02:05 --> 00:02:11
The problem set has y prime
equals, let's see,
33
00:02:11 --> 00:02:17
x minus y squared.
34
00:02:15 --> 00:02:21
And, it also has y prime equals
y minus x squared.
35
00:02:22 --> 00:02:28
There are others,
36
00:02:25 --> 00:02:31
too.
Now, when you look at this,
37
00:02:29 --> 00:02:35
this, of course,
you can solve by separating
38
00:02:35 --> 00:02:41
variables.
So, this is solvable.
39
00:02:39 --> 00:02:45
This one is-- and neither of
these can you separate
40
00:02:43 --> 00:02:49
variables.
And they look extremely
41
00:02:46 --> 00:02:52
similar.
But they are extremely
42
00:02:48 --> 00:02:54
dissimilar.
The most dissimilar about them
43
00:02:52 --> 00:02:58
is that this one is easily
solvable.
44
00:02:54 --> 00:03:00
And you will learn,
if you don't know already,
45
00:02:58 --> 00:03:04
next time next Friday how to
solve this one.
46
00:03:03 --> 00:03:09
This one, which looks almost
the same, is unsolvable in a
47
00:03:06 --> 00:03:12
certain sense.
Namely, there are no elementary
48
00:03:09 --> 00:03:15
functions which you can write
down, which will give a solution
49
00:03:13 --> 00:03:19
of that differential equation.
So, right away,
50
00:03:16 --> 00:03:22
one confronts the most
significant fact that even for
51
00:03:19 --> 00:03:25
the simplest possible
differential equations,
52
00:03:22 --> 00:03:28
those which only involve the
first derivative,
53
00:03:25 --> 00:03:31
it's possible to write down
extremely looking simple guys.
54
00:03:30 --> 00:03:36
I'll put this one up in blue to
indicate that it's bad.
55
00:03:35 --> 00:03:41
Whoops, sorry,
I mean, not really bad,
56
00:03:38 --> 00:03:44
but recalcitrant.
It's not solvable in the
57
00:03:42 --> 00:03:48
ordinary sense in which you
think of an equation is
58
00:03:46 --> 00:03:52
solvable.
And, since those equations are
59
00:03:50 --> 00:03:56
the rule rather than the
exception, I'm going about this
60
00:03:55 --> 00:04:01
first day to not solving a
single differential equation,
61
00:04:00 --> 00:04:06
but indicating to you what you
do when you meet a blue equation
62
00:04:06 --> 00:04:12
like that.
What do you do with it?
63
00:04:11 --> 00:04:17
So, this first day is going to
be devoted to geometric ways of
64
00:04:17 --> 00:04:23
looking at differential
equations and numerical.
65
00:04:21 --> 00:04:27
At the very end,
I'll talk a little bit about
66
00:04:25 --> 00:04:31
numerical ways.
And you'll work on both of
67
00:04:29 --> 00:04:35
those for the first problem set.
So, what's our geometric view
68
00:04:35 --> 00:04:41
of differential equations?
Well, it's something that's
69
00:04:41 --> 00:04:47
contrasted with the usual
procedures, by which you solve
70
00:04:45 --> 00:04:51
things and find elementary
functions which solve them.
71
00:04:49 --> 00:04:55
I'll call that the analytic
method.
72
00:04:52 --> 00:04:58
So, on the one hand,
we have the analytic ideas,
73
00:04:56 --> 00:05:02
in which you write down
explicitly the equation,
74
00:04:59 --> 00:05:05
y prime equals f of x,y.
75
00:05:04 --> 00:05:10
And, you look for certain
functions, which are called its
76
00:05:07 --> 00:05:13
solutions.
Now, so there's the ODE.
77
00:05:09 --> 00:05:15
And, y1 of x,
notice I don't use a separate
78
00:05:12 --> 00:05:18
letter.
I don't use g or h or something
79
00:05:14 --> 00:05:20
like that for the solution
because the letters multiply so
80
00:05:18 --> 00:05:24
quickly, that is,
multiply in the sense of
81
00:05:20 --> 00:05:26
rabbits, that after a while,
if you keep using different
82
00:05:24 --> 00:05:30
letters for each new idea,
you can't figure out what
83
00:05:27 --> 00:05:33
you're talking about.
So, I'll use y1 means,
84
00:05:32 --> 00:05:38
it's a solution of this
differential equation.
85
00:05:37 --> 00:05:43
Of course, the differential
equation has many solutions
86
00:05:43 --> 00:05:49
containing an arbitrary
constant.
87
00:05:46 --> 00:05:52
So, we'll call this the
solution.
88
00:05:50 --> 00:05:56
Now, the geometric view,
the geometric guy that
89
00:05:54 --> 00:06:00
corresponds to this version of
writing the equation,
90
00:06:00 --> 00:06:06
is something called a direction
field.
91
00:06:06 --> 00:06:12
And, the solution is,
from the geometric point of
92
00:06:09 --> 00:06:15
view, something called an
integral curve.
93
00:06:12 --> 00:06:18
So, let me explain if you don't
know what the direction field
94
00:06:16 --> 00:06:22
is.
I know for some of you,
95
00:06:18 --> 00:06:24
I'm reviewing what you learned
in high school.
96
00:06:21 --> 00:06:27
Those of you who had the BC
syllabus in high school should
97
00:06:25 --> 00:06:31
know these things.
But, it never hurts to get a
98
00:06:28 --> 00:06:34
little more practice.
And, in any event,
99
00:06:31 --> 00:06:37
I think the computer stuff that
you will be doing on the problem
100
00:06:36 --> 00:06:42
set, a certain amount of it
should be novel to you.
101
00:06:41 --> 00:06:47
It was novel to me,
so why not to you?
102
00:06:43 --> 00:06:49
So, what's a direction field?
Well, the direction field is,
103
00:06:47 --> 00:06:53
you take the plane,
and in each point of the
104
00:06:51 --> 00:06:57
plane-- of course,
that's an impossibility.
105
00:06:54 --> 00:07:00
But, you pick some points of
the plane.
106
00:06:56 --> 00:07:02
You draw what's called a little
line element.
107
00:07:01 --> 00:07:07
So, there is a point.
It's a little line,
108
00:07:04 --> 00:07:10
and the only thing which
distinguishes it outside of its
109
00:07:08 --> 00:07:14
position in the plane,
so here's the point,
110
00:07:11 --> 00:07:17
(x,y), at which we are drawing
this line element,
111
00:07:15 --> 00:07:21
is its slope.
And, what is its slope?
112
00:07:18 --> 00:07:24
Its slope is to be f of x,y.
113
00:07:21 --> 00:07:27
And now, You fill up the plane
with these things until you're
114
00:07:26 --> 00:07:32
tired of putting then in.
So, I'm going to get tired
115
00:07:30 --> 00:07:36
pretty quickly.
So, I don't know,
116
00:07:34 --> 00:07:40
let's not make them all go the
same way.
117
00:07:36 --> 00:07:42
That sort of seems cheating.
How about here?
118
00:07:40 --> 00:07:46
Here's a few randomly chosen
line elements that I put in,
119
00:07:44 --> 00:07:50
and I putted the slopes at
random since I didn't have any
120
00:07:48 --> 00:07:54
particular differential equation
in mind.
121
00:07:50 --> 00:07:56
Now, the integral curve,
so those are the line elements.
122
00:07:54 --> 00:08:00
The integral curve is a curve,
which goes through the plane,
123
00:07:58 --> 00:08:04
and at every point is tangent
to the line element there.
124
00:08:04 --> 00:08:10
So, this is the integral curve.
Hey, wait a minute,
125
00:08:07 --> 00:08:13
I thought tangents were the
line element there didn't even
126
00:08:12 --> 00:08:18
touch it.
Well, I can't fill up the plane
127
00:08:15 --> 00:08:21
with line elements.
Here, at this point,
128
00:08:17 --> 00:08:23
there was a line element,
which I didn't bother drawing
129
00:08:22 --> 00:08:28
in.
And, it was tangent to that.
130
00:08:24 --> 00:08:30
Same thing over here:
if I drew the line element
131
00:08:27 --> 00:08:33
here, I would find that the
curve had exactly the right
132
00:08:31 --> 00:08:37
slope there.
So, the point is the integral,
133
00:08:37 --> 00:08:43
what distinguishes the integral
curve is that everywhere it has
134
00:08:43 --> 00:08:49
the direction,
that's the way I'll indicate
135
00:08:47 --> 00:08:53
that it's tangent,
has the direction of the field
136
00:08:52 --> 00:08:58
everywhere at all points on the
curve, of course,
137
00:08:57 --> 00:09:03
where it doesn't go.
It doesn't have any mission to
138
00:09:02 --> 00:09:08
fulfill.
Now, I say that this integral
139
00:09:04 --> 00:09:10
curve is the graph of the
solution to the differential
140
00:09:08 --> 00:09:14
equation.
In other words,
141
00:09:10 --> 00:09:16
writing down analytically the
differential equation is the
142
00:09:14 --> 00:09:20
same geometrically as drawing
this direction field,
143
00:09:18 --> 00:09:24
and solving analytically for a
solution of the differential
144
00:09:22 --> 00:09:28
equation is the same thing as
geometrically drawing an
145
00:09:26 --> 00:09:32
integral curve.
So, what am I saying?
146
00:09:30 --> 00:09:36
I say that an integral curve,
all right, let me write it this
147
00:09:39 --> 00:09:45
way.
I'll make a little theorem out
148
00:09:44 --> 00:09:50
of it, that y1 of x is
a solution to the differential
149
00:09:53 --> 00:09:59
equation if, and only if,
the graph, the curve associated
150
00:10:01 --> 00:10:07
with this, the graph of y1 of x
is an integral curve.
151
00:10:11 --> 00:10:17
Integral curve of what?
Well, of the direction field
152
00:10:14 --> 00:10:20
associated with that equation.
But there isn't quite enough
153
00:10:18 --> 00:10:24
room to write that on the board.
But, you could put it in your
154
00:10:22 --> 00:10:28
notes, if you take notes.
So, this is the relation
155
00:10:25 --> 00:10:31
between the two,
the integral curves of the
156
00:10:28 --> 00:10:34
graphs or solutions.
Now, why is that so?
157
00:10:31 --> 00:10:37
Well, in fact,
all I have to do to prove this,
158
00:10:34 --> 00:10:40
if you can call it a proof at
all, is simply to translate what
159
00:10:38 --> 00:10:44
each side really means.
What does it really mean to say
160
00:10:42 --> 00:10:48
that a given function is a
solution to the differential
161
00:10:45 --> 00:10:51
equation?
Well, it means that if you plug
162
00:10:48 --> 00:10:54
it into the differential
equation, it satisfies it.
163
00:10:52 --> 00:10:58
Okay, what is that?
So, how do I plug it into the
164
00:10:55 --> 00:11:01
differential equation and check
that it satisfies it?
165
00:11:00 --> 00:11:06
Well, doing it in the abstract,
I first calculate its
166
00:11:04 --> 00:11:10
derivative.
And then, how will it look
167
00:11:07 --> 00:11:13
after I plugged it into the
differential equation?
168
00:11:12 --> 00:11:18
Well, I don't do anything to
the x, but wherever I see y,
169
00:11:17 --> 00:11:23
I plug in this particular
function.
170
00:11:20 --> 00:11:26
So, in notation,
that would be written this way.
171
00:11:24 --> 00:11:30
So, for this to be a solution
means this, that that equation
172
00:11:29 --> 00:11:35
is satisfied.
Okay, what does it mean for the
173
00:11:35 --> 00:11:41
graph to be an integral curve?
Well, it means that at each
174
00:11:42 --> 00:11:48
point, the slope of this curve,
it means that the slope of y1
175
00:11:49 --> 00:11:55
of x should be,
at each point, x1 y1.
176
00:11:52 --> 00:11:58
It should be equal to the slope
177
00:11:58 --> 00:12:04
of the direction field at that
point.
178
00:12:04 --> 00:12:10
And then, what is the slope of
the direction field at that
179
00:12:08 --> 00:12:14
point?
Well, it is f of that
180
00:12:10 --> 00:12:16
particular, well,
at the point,
181
00:12:12 --> 00:12:18
x, y1 of x.
If you like,
182
00:12:15 --> 00:12:21
you can put a subscript,
one, on there,
183
00:12:18 --> 00:12:24
send a one here or a zero
there, to indicate that you mean
184
00:12:22 --> 00:12:28
a particular point.
But, it looks better if you
185
00:12:26 --> 00:12:32
don't.
But, there's some possibility
186
00:12:28 --> 00:12:34
of confusion.
I admit to that.
187
00:12:32 --> 00:12:38
So, the slope of the direction
field, what is that slope?
188
00:12:35 --> 00:12:41
Well, by the way,
I calculated the direction
189
00:12:38 --> 00:12:44
field.
Its slope at the point was to
190
00:12:41 --> 00:12:47
be x, whatever the value of x
was, and whatever the value of
191
00:12:45 --> 00:12:51
y1 of x was,
substituted into the right-hand
192
00:12:49 --> 00:12:55
side of the equation.
So, what the slope of this
193
00:12:52 --> 00:12:58
function of that curve of the
graph should be equal to the
194
00:12:56 --> 00:13:02
slope of the direction field.
Now, what does this say?
195
00:13:01 --> 00:13:07
Well, what's the slope of y1 of
x?
196
00:13:03 --> 00:13:09
That's y1 prime of x.
197
00:13:05 --> 00:13:11
That's from the first day of
18.01, calculus.
198
00:13:08 --> 00:13:14
What's the slope of the
direction field?
199
00:13:11 --> 00:13:17
This?
Well, it's this.
200
00:13:12 --> 00:13:18
And, that's with the right hand
side.
201
00:13:14 --> 00:13:20
So, saying these two guys are
the same or equal,
202
00:13:17 --> 00:13:23
is exactly, analytically,
the same as saying these two
203
00:13:21 --> 00:13:27
guys are equal.
So, in other words,
204
00:13:23 --> 00:13:29
the proof consists of,
what does this really mean?
205
00:13:26 --> 00:13:32
What does this really mean?
And after you see what both
206
00:13:29 --> 00:13:35
really mean, you say,
yeah, they're the same.
207
00:13:34 --> 00:13:40
So, I don't how to write that.
It's okay: same,
208
00:13:39 --> 00:13:45
same, how's that?
This is the same as that.
209
00:13:44 --> 00:13:50
Okay, well, this leaves us the
interesting question of how do
210
00:13:52 --> 00:13:58
you draw a direction from the,
well, this being 2003,
211
00:13:58 --> 00:14:04
mostly computers draw them for
you.
212
00:14:04 --> 00:14:10
Nonetheless,
you do have to know a certain
213
00:14:07 --> 00:14:13
amount.
I've given you a couple of
214
00:14:09 --> 00:14:15
exercises where you have to draw
the direction field yourself.
215
00:14:14 --> 00:14:20
This is so you get a feeling
for it, and also because humans
216
00:14:19 --> 00:14:25
don't draw direction fields the
same way computers do.
217
00:14:23 --> 00:14:29
So, let's first of all,
how did computers do it?
218
00:14:27 --> 00:14:33
They are very stupid.
There's no problem.
219
00:14:32 --> 00:14:38
Since they go very fast and
have unlimited amounts of energy
220
00:14:37 --> 00:14:43
to waste, the computer method is
the naive one.
221
00:14:42 --> 00:14:48
You pick the point.
You pick a point,
222
00:14:45 --> 00:14:51
and generally,
they are usually equally
223
00:14:49 --> 00:14:55
spaced.
You determine some spacing,
224
00:14:52 --> 00:14:58
that one: blah,
blah, blah, blah,
225
00:14:55 --> 00:15:01
blah, blah, blah,
equally spaced.
226
00:15:00 --> 00:15:06
And, at each point,
it computes f of x,
227
00:15:04 --> 00:15:10
y at the point,
finds, meets,
228
00:15:08 --> 00:15:14
and computes the value of f of
(x, y), that function,
229
00:15:14 --> 00:15:20
and the next thing is,
on the screen,
230
00:15:17 --> 00:15:23
it draws, at (x,
y), the little line element
231
00:15:22 --> 00:15:28
having slope f of x,y.
232
00:15:26 --> 00:15:32
In other words,
it does what the differential
233
00:15:30 --> 00:15:36
equation tells it to do.
And the only thing that it does
234
00:15:36 --> 00:15:42
is you can, if you are telling
the thing to draw the direction
235
00:15:40 --> 00:15:46
field, about the only option you
have is telling what the spacing
236
00:15:43 --> 00:15:49
should be, and sometimes people
don't like to see a whole line.
237
00:15:46 --> 00:15:52
They only like to see a little
bit of a half line.
238
00:15:49 --> 00:15:55
And, you can sometimes tell,
according to the program,
239
00:15:52 --> 00:15:58
tell the computer how long you
want that line to be,
240
00:15:55 --> 00:16:01
if you want it teeny or a
little bigger.
241
00:15:57 --> 00:16:03
Once in awhile you want you
want it narrower on it,
242
00:16:00 --> 00:16:06
but not right now.
Okay, that's what a computer
243
00:16:04 --> 00:16:10
does.
What does a human do?
244
00:16:05 --> 00:16:11
This is what it means to be
human.
245
00:16:08 --> 00:16:14
You use your intelligence.
From a human point of view,
246
00:16:12 --> 00:16:18
this stuff has been done in the
wrong order.
247
00:16:15 --> 00:16:21
And the reason it's been done
in the wrong order:
248
00:16:18 --> 00:16:24
because for each new point,
it requires a recalculation of
249
00:16:22 --> 00:16:28
f of (x, y).
250
00:16:24 --> 00:16:30
That is horrible.
The computer doesn't mind,
251
00:16:27 --> 00:16:33
but a human does.
So, for a human,
252
00:16:31 --> 00:16:37
the way to do it is not to
begin by picking the point,
253
00:16:35 --> 00:16:41
but to begin by picking the
slope that you would like to
254
00:16:40 --> 00:16:46
see.
So, you begin by taking the
255
00:16:42 --> 00:16:48
slope.
Let's call it the value of the
256
00:16:45 --> 00:16:51
slope, C.
So, you pick a number.
257
00:16:48 --> 00:16:54
C is two.
I want to see where are all the
258
00:16:51 --> 00:16:57
points in the plane where the
slope of that line element would
259
00:16:56 --> 00:17:02
be two?
Well, they will satisfy an
260
00:16:58 --> 00:17:04
equation.
The equation is f of (x,
261
00:17:02 --> 00:17:08
y) equals, in general,
it will be C.
262
00:17:07 --> 00:17:13
So, what you do is plot this,
plot the equation,
263
00:17:10 --> 00:17:16
plot this equation.
Notice, it's not the
264
00:17:14 --> 00:17:20
differential equation.
You can't exactly plot a
265
00:17:17 --> 00:17:23
differential equation.
It's a curve,
266
00:17:20 --> 00:17:26
an ordinary curve.
But which curve will depend;
267
00:17:24 --> 00:17:30
it's, in fact,
from the 18.02 point of view,
268
00:17:28 --> 00:17:34
the level curve of C,
sorry, it's a level curve of f
269
00:17:32 --> 00:17:38
of (x, y), the function f of x
and y corresponding to the level
270
00:17:37 --> 00:17:43
of value C.
But we are not going to call it
271
00:17:42 --> 00:17:48
that because this is not 18.02.
Instead, we're going to call it
272
00:17:48 --> 00:17:54
an isocline.
And then, you plot,
273
00:17:51 --> 00:17:57
well, you've done it.
So, you've got this isocline,
274
00:17:56 --> 00:18:02
except I'm going to use a
solution curve,
275
00:18:00 --> 00:18:06
solid lines,
only for integral curves.
276
00:18:03 --> 00:18:09
When we do plot isoclines,
to indicate that they are not
277
00:18:09 --> 00:18:15
solutions, we'll use dashed
lines for doing them.
278
00:18:15 --> 00:18:21
One of the computer things does
and the other one doesn't.
279
00:18:18 --> 00:18:24
But they use different colors,
also.
280
00:18:20 --> 00:18:26
There are different ways of
telling you what's an isocline
281
00:18:23 --> 00:18:29
and what's the solution curve.
So, and what do you do?
282
00:18:26 --> 00:18:32
So, these are all the points
where the slope is going to be
283
00:18:29 --> 00:18:35
C.
And now, what you do is draw in
284
00:18:32 --> 00:18:38
as many as you want of line
elements having slope C.
285
00:18:35 --> 00:18:41
Notice how efficient that is.
If you want 50 million of them
286
00:18:39 --> 00:18:45
and have the time,
draw in 50 million.
287
00:18:41 --> 00:18:47
If two or three are enough,
draw in two or three.
288
00:18:45 --> 00:18:51
You will be looking at the
picture.
289
00:18:47 --> 00:18:53
You will see what the curve
looks like, and that will give
290
00:18:51 --> 00:18:57
you your judgment as to how you
are to do that.
291
00:18:54 --> 00:19:00
So, in general,
a picture drawn that way,
292
00:18:57 --> 00:19:03
so let's say,
an isocline corresponding to C
293
00:18:59 --> 00:19:05
equals zero.
The line elements,
294
00:19:03 --> 00:19:09
and I think for an isocline,
for the purposes of this
295
00:19:07 --> 00:19:13
lecture, it would be a good idea
to put isoclines.
296
00:19:10 --> 00:19:16
Okay, so I'm going to put
solution curves in pink,
297
00:19:14 --> 00:19:20
or whatever this color is,
and isoclines are going to be
298
00:19:18 --> 00:19:24
in orange, I guess.
So, isocline,
299
00:19:21 --> 00:19:27
represented by a dashed line,
and now you will put in the
300
00:19:25 --> 00:19:31
line elements of,
we'll need lots of chalk for
301
00:19:28 --> 00:19:34
that.
So, I'll use white chalk.
302
00:19:32 --> 00:19:38
Y horizontal?
Because according to this the
303
00:19:34 --> 00:19:40
slope is supposed to be zero
there.
304
00:19:37 --> 00:19:43
And at the same way,
how about an isocline where the
305
00:19:40 --> 00:19:46
slope is negative one?
Let's suppose here C is equal
306
00:19:44 --> 00:19:50
to negative one.
Okay, then it will look like
307
00:19:47 --> 00:19:53
this.
These are supposed to be lines
308
00:19:49 --> 00:19:55
of slope negative one.
Don't shoot me if they are not.
309
00:19:53 --> 00:19:59
So, that's the principle.
So, this is how you will fill
310
00:19:56 --> 00:20:02
up the plane to draw a direction
field: by plotting the isoclines
311
00:20:01 --> 00:20:07
first.
And then, once you have the
312
00:20:04 --> 00:20:10
isoclines there,
you will have line elements.
313
00:20:07 --> 00:20:13
And you can draw a direction
field.
314
00:20:09 --> 00:20:15
Okay, so, for the next few
minutes, I'd like to work a
315
00:20:12 --> 00:20:18
couple of examples for you to
show how this works out in
316
00:20:15 --> 00:20:21
practice.
317
00:20:17 --> 00:20:23
318
319
320
00:20:34 --> 00:20:40
So, the first equation is going
to be y prime equals minus x
321
00:20:45 --> 00:20:51
over y.
Okay, first thing,
322
00:20:53 --> 00:20:59
what are the isoclines?
Well, the isoclines are going
323
00:21:03 --> 00:21:09
to be y.
Well, negative x over y is
324
00:21:08 --> 00:21:14
equal to C.
Maybe I better make two steps
325
00:21:12 --> 00:21:18
out of this.
Minus x over y is equal to C.
326
00:21:16 --> 00:21:22
But, of course,
nobody draws a curve in that
327
00:21:19 --> 00:21:25
form.
You'll want it in the form y
328
00:21:22 --> 00:21:28
equals minus one over
C times x.
329
00:21:26 --> 00:21:32
So, there's our isocline.
Why don't I put that up in
330
00:21:32 --> 00:21:38
orange since it's going to be,
that's the color I'll draw it
331
00:21:36 --> 00:21:42
in.
In other words,
332
00:21:38 --> 00:21:44
for different values of C,
now this thing is aligned.
333
00:21:42 --> 00:21:48
It's aligned,
in fact, through the origin.
334
00:21:45 --> 00:21:51
This looks pretty simple.
Okay, so here's our plane.
335
00:21:50 --> 00:21:56
The isoclines are going to be
lines through the origin.
336
00:21:54 --> 00:22:00
And now, let's put them in,
suppose, for example,
337
00:21:58 --> 00:22:04
C is equal to one.
Well, if C is equal to one,
338
00:22:06 --> 00:22:12
then it's the line,
y equals minus x.
339
00:22:14 --> 00:22:20
So, this is the isocline.
I'll put, down here,
340
00:22:23 --> 00:22:29
C equals minus one.
And, along it,
341
00:22:30 --> 00:22:36
no, something's wrong.
I'm sorry?
342
00:22:38 --> 00:22:44
C is one, not negative one,
right, thanks.
343
00:22:42 --> 00:22:48
Thanks.
So, C equals one.
344
00:22:44 --> 00:22:50
So, it should be little line
segments of slope one will be
345
00:22:50 --> 00:22:56
the line elements,
things of slope one.
346
00:22:54 --> 00:23:00
OK, now how about C equals
negative one?
347
00:23:00 --> 00:23:06
If C equals negative one,
then it's the line,
348
00:23:03 --> 00:23:09
y equals x.
And so, that's the isocline.
349
00:23:07 --> 00:23:13
Notice, still dash because
these are isoclines.
350
00:23:11 --> 00:23:17
Here, C is negative one.
And so, the slope elements look
351
00:23:15 --> 00:23:21
like this.
Notice, they are perpendicular.
352
00:23:19 --> 00:23:25
Now, notice that they are
always going to be perpendicular
353
00:23:23 --> 00:23:29
to the line because the slope of
this line is minus one over C.
354
00:23:30 --> 00:23:36
But, the slope of the line
element is going to be C.
355
00:23:33 --> 00:23:39
Those numbers,
minus one over C and C,
356
00:23:36 --> 00:23:42
are negative reciprocals.
And, you know that two lines
357
00:23:40 --> 00:23:46
whose slopes are negative
reciprocals are perpendicular.
358
00:23:44 --> 00:23:50
So, the line elements are going
to be perpendicular to these.
359
00:23:49 --> 00:23:55
And therefore,
I hardly even have to bother
360
00:23:52 --> 00:23:58
calculating, doing any more
calculation.
361
00:23:55 --> 00:24:01
Here's going to be a,
well, how about this one?
362
00:24:00 --> 00:24:06
Here's a controversial
isocline.
363
00:24:02 --> 00:24:08
Is that an isocline?
Well, wait a minute.
364
00:24:05 --> 00:24:11
That doesn't correspond to
anything looking like this.
365
00:24:10 --> 00:24:16
Ah-ha, but it would if I put C
multiplied through by C.
366
00:24:14 --> 00:24:20
And then, it would correspond
to C being zero.
367
00:24:18 --> 00:24:24
In other words,
don't write it like this.
368
00:24:21 --> 00:24:27
Multiply through by C.
It will read C y equals
369
00:24:25 --> 00:24:31
negative x.
And then, when C is zero,
370
00:24:29 --> 00:24:35
I have x equals zero,
which is exactly the y-axis.
371
00:24:35 --> 00:24:41
So, that really is included.
How about the x-axis?
372
00:24:38 --> 00:24:44
Well, the x-axis is not
included.
373
00:24:40 --> 00:24:46
However, most people include it
anyway.
374
00:24:43 --> 00:24:49
This is very common to be a
sort of sloppy and bending the
375
00:24:47 --> 00:24:53
edges of corners a little bit,
and hoping nobody will notice.
376
00:24:51 --> 00:24:57
We'll say that corresponds to C
equals infinity.
377
00:24:55 --> 00:25:01
I hope nobody wants to fight
about that.
378
00:24:58 --> 00:25:04
If you do, go fight with
somebody else.
379
00:25:02 --> 00:25:08
So, if C is infinity,
that means the little line
380
00:25:05 --> 00:25:11
segment should have infinite
slope, and by common consent,
381
00:25:10 --> 00:25:16
that means it should be
vertical.
382
00:25:12 --> 00:25:18
And so, we can even count this
as sort of an isocline.
383
00:25:17 --> 00:25:23
And, I'll make the dashes
smaller, indicate it has a lower
384
00:25:21 --> 00:25:27
status than the others.
And, I'll put this in,
385
00:25:25 --> 00:25:31
do this weaselly thing of
putting it in quotation marks to
386
00:25:29 --> 00:25:35
indicate that I'm not
responsible for it.
387
00:25:34 --> 00:25:40
Okay, now, we now have to put
it the integral curves.
388
00:25:39 --> 00:25:45
Well, nothing could be easier.
I'm looking for curves which
389
00:25:45 --> 00:25:51
are everywhere perpendicular to
these rays.
390
00:25:50 --> 00:25:56
Well, you know from geometry
that those are circles.
391
00:25:55 --> 00:26:01
So, the integral curves are
circles.
392
00:26:00 --> 00:26:06
And, it's an elementary
exercise, which I would not
393
00:26:04 --> 00:26:10
deprive you of the pleasure of.
Solve the ODE by separation of
394
00:26:08 --> 00:26:14
variables.
In other words,
395
00:26:10 --> 00:26:16
we've gotten the,
so the circles are ones with a
396
00:26:14 --> 00:26:20
center at the origin,
of course, equal some constant.
397
00:26:18 --> 00:26:24
I'll call it C1,
so it's not confused with this
398
00:26:22 --> 00:26:28
C.
They look like that,
399
00:26:24 --> 00:26:30
and now you should solve this
by separating variables,
400
00:26:28 --> 00:26:34
and just confirm that the
solutions are,
401
00:26:31 --> 00:26:37
in fact, those circles.
One interesting thing,
402
00:26:36 --> 00:26:42
and so I confirm this,
I won't do it because I want to
403
00:26:40 --> 00:26:46
do geometric and numerical
things today.
404
00:26:42 --> 00:26:48
So, if you solve it by
separating variables,
405
00:26:45 --> 00:26:51
one interesting thing to note
is that if I write the solution
406
00:26:49 --> 00:26:55
as y equals y1 of x, well,
407
00:26:52 --> 00:26:58
it'll look something like the
square root of C1 minus,
408
00:26:56 --> 00:27:02
let's make this squared because
that's the way people usually
409
00:27:00 --> 00:27:06
put the radius,
minus x squared.
410
00:27:03 --> 00:27:09
And so, a solution,
411
00:27:06 --> 00:27:12
a typical solution looks like
this.
412
00:27:09 --> 00:27:15
Well, what's the solution over
here?
413
00:27:11 --> 00:27:17
Well, that one solution will be
goes from here to here.
414
00:27:15 --> 00:27:21
If you like,
it has a negative side to it.
415
00:27:18 --> 00:27:24
So, I'll make,
let's say, plus.
416
00:27:21 --> 00:27:27
There's another solution,
which has a negative value.
417
00:27:25 --> 00:27:31
But let's use the one with the
positive value of the square
418
00:27:29 --> 00:27:35
root.
My point is this,
419
00:27:32 --> 00:27:38
that that solution,
the domain of that solution,
420
00:27:35 --> 00:27:41
really only goes from here to
here.
421
00:27:38 --> 00:27:44
It's not the whole x-axis.
It's just a limited piece of
422
00:27:42 --> 00:27:48
the x-axis where that solution
is defined.
423
00:27:45 --> 00:27:51
There's no way of extending it
further.
424
00:27:48 --> 00:27:54
And, there's no way of
predicting, by looking at the
425
00:27:52 --> 00:27:58
differential equation,
that a typical solution was
426
00:27:56 --> 00:28:02
going to have a limited domain
like that.
427
00:28:01 --> 00:28:07
In other words,
you could find a solution,
428
00:28:04 --> 00:28:10
but how far out is it going to
go?
429
00:28:07 --> 00:28:13
Sometimes, it's impossible to
tell, except by either finding
430
00:28:12 --> 00:28:18
it explicitly,
or by asking a computer to draw
431
00:28:16 --> 00:28:22
a picture of it,
and seeing if that gives you
432
00:28:19 --> 00:28:25
some insight.
It's one of the many
433
00:28:22 --> 00:28:28
difficulties in handling
differential equations.
434
00:28:26 --> 00:28:32
You don't know what the domain
of a solution is going to be
435
00:28:31 --> 00:28:37
until you've actually calculated
it.
436
00:28:36 --> 00:28:42
Now, a slightly more
complicated example is going to
437
00:28:40 --> 00:28:46
be, let's see, y prime
equals one plus x minus y.
438
00:28:43 --> 00:28:49
It's not a lot more
439
00:28:46 --> 00:28:52
complicated, and as a computer
exercise, you will work with,
440
00:28:51 --> 00:28:57
still, more complicated ones.
But here, the isoclines would
441
00:28:56 --> 00:29:02
be what?
Well, I set that equal to C.
442
00:29:00 --> 00:29:06
Can you do the algebra in your
head?
443
00:29:02 --> 00:29:08
An isocline will have the
equation: this equals C.
444
00:29:07 --> 00:29:13
So, I'm going to put the y on
the right hand side,
445
00:29:11 --> 00:29:17
and that C on the left hand
side.
446
00:29:13 --> 00:29:19
So, it will have the equation y
equals one plus x minus C,
447
00:29:19 --> 00:29:25
or a nicer way to
write it would be x plus one
448
00:29:23 --> 00:29:29
minus C.
I guess it really doesn't
449
00:29:28 --> 00:29:34
matter.
So there's the equation of the
450
00:29:31 --> 00:29:37
isocline.
Let's quickly draw the
451
00:29:34 --> 00:29:40
direction field.
And notice, by the way,
452
00:29:36 --> 00:29:42
it's a simple equation,
but you cannot separate
453
00:29:39 --> 00:29:45
variables.
So, I will not,
454
00:29:41 --> 00:29:47
today at any rate,
be able to check the answer.
455
00:29:44 --> 00:29:50
I will not be able to get an
analytic answer.
456
00:29:47 --> 00:29:53
All we'll be able to do now is
get a geometric answer.
457
00:29:50 --> 00:29:56
But notice how quickly,
relatively quickly,
458
00:29:53 --> 00:29:59
one can get it.
So, I'm feeling for how the
459
00:29:56 --> 00:30:02
solutions behave to this
equation.
460
00:30:00 --> 00:30:06
All right, let's see,
what should we plot first?
461
00:30:05 --> 00:30:11
I like C equals one,
no, don't do C equals one.
462
00:30:10 --> 00:30:16
Let's do C equals zero,
first.
463
00:30:13 --> 00:30:19
C equals zero.
That's the line.
464
00:30:16 --> 00:30:22
y equals x plus 1.
465
00:30:19 --> 00:30:25
Okay, let me run and get that
chalk.
466
00:30:23 --> 00:30:29
So, I'll isoclines are in
orange.
467
00:30:27 --> 00:30:33
If so, when C equals zero,
y equals x plus one.
468
00:30:32 --> 00:30:38
So, let's say it's this curve.
C equals zero.
469
00:30:38 --> 00:30:44
How about C equals negative
one?
470
00:30:42 --> 00:30:48
Then it's y equals x plus two.
471
00:30:47 --> 00:30:53
It's this curve.
Well, let's label it down here.
472
00:30:53 --> 00:30:59
So, this is C equals negative
one.
473
00:30:57 --> 00:31:03
C equals negative two would be
y equals x, no,
474
00:31:02 --> 00:31:08
what am I doing?
C equals negative one is y
475
00:31:08 --> 00:31:14
equals x plus two.
That's right.
476
00:31:12 --> 00:31:18
Well, how about the other side?
If C equals plus one,
477
00:31:16 --> 00:31:22
well, then it's going to go
through the origin.
478
00:31:20 --> 00:31:26
It looks like a little more
room down here.
479
00:31:24 --> 00:31:30
How about, so if this is going
to be C equals one,
480
00:31:28 --> 00:31:34
then I sort of get the idea.
C equals two will look like
481
00:31:34 --> 00:31:40
this.
They're all going to be
482
00:31:37 --> 00:31:43
parallel lines because all
that's changing is the
483
00:31:42 --> 00:31:48
y-intercept, as I do this thing.
So, here, it's C equals two.
484
00:31:47 --> 00:31:53
That's probably enough.
All right, let's put it in the
485
00:31:53 --> 00:31:59
line elements.
All right, C equals negative
486
00:31:57 --> 00:32:03
one.
These will be perpendicular.
487
00:32:00 --> 00:32:06
C equals zero,
like this.
488
00:32:04 --> 00:32:10
C equals one.
Oh, this is interesting.
489
00:32:06 --> 00:32:12
I can't even draw in the line
elements because they seem to
490
00:32:10 --> 00:32:16
coincide with the curve itself,
with the line itself.
491
00:32:14 --> 00:32:20
They write y along the line,
and that makes it hard to draw
492
00:32:18 --> 00:32:24
them in.
How about C equals two?
493
00:32:20 --> 00:32:26
Well, here, the line elements
will be slanty.
494
00:32:23 --> 00:32:29
They'll have slope two,
so a pretty slanty up.
495
00:32:26 --> 00:32:32
And, I can see if a C equals
three in the same way.
496
00:32:31 --> 00:32:37
There are going to be even more
slantier up.
497
00:32:34 --> 00:32:40
And here, they're going to be
even more slanty down.
498
00:32:37 --> 00:32:43
This is not very scientific
terminology or mathematical,
499
00:32:41 --> 00:32:47
but you get the idea.
Okay, so there's our quick
500
00:32:45 --> 00:32:51
version of the direction field.
All we have to do is put in
501
00:32:49 --> 00:32:55
some integral curves now.
Well, it looks like it's doing
502
00:32:53 --> 00:32:59
this.
It gets less slanty here.
503
00:32:55 --> 00:33:01
It levels out,
has slope zero.
504
00:32:59 --> 00:33:05
And now, in this part of the
plain, the slope seems to be
505
00:33:03 --> 00:33:09
rising.
So, it must do something like
506
00:33:06 --> 00:33:12
that.
This guy must do something like
507
00:33:08 --> 00:33:14
this.
I'm a little doubtful of what I
508
00:33:11 --> 00:33:17
should be doing here.
Or, how about going from the
509
00:33:15 --> 00:33:21
other side?
Well, it rises,
510
00:33:17 --> 00:33:23
gets a little,
should it cross this?
511
00:33:20 --> 00:33:26
What should I do?
Well, there's one integral
512
00:33:23 --> 00:33:29
curve, which is easy to see.
It's this one.
513
00:33:26 --> 00:33:32
This line is both an isocline
and an integral curve.
514
00:33:32 --> 00:33:38
It's everything,
except drawable,
515
00:33:35 --> 00:33:41
[LAUGHTER] so,
you understand this is the same
516
00:33:41 --> 00:33:47
line.
It's both orange and pink at
517
00:33:45 --> 00:33:51
the same time.
But I don't know what
518
00:33:49 --> 00:33:55
combination color that would
make.
519
00:33:53 --> 00:33:59
It doesn't look like a line,
but be sympathetic.
520
00:34:00 --> 00:34:06
Now, the question is,
what's happening in this
521
00:34:04 --> 00:34:10
corridor?
In the corridor,
522
00:34:06 --> 00:34:12
that's not a mathematical word
either, between the isoclines
523
00:34:12 --> 00:34:18
for, well, what are they?
They are the isoclines for C
524
00:34:18 --> 00:34:24
equals two, and C equals zero.
How does that corridor look?
525
00:34:23 --> 00:34:29
Well: something like this.
Over here, the lines all look
526
00:34:29 --> 00:34:35
like that.
And here, they all look like
527
00:34:33 --> 00:34:39
this.
The slope is two.
528
00:34:36 --> 00:34:42
And, a hapless solution gets in
there.
529
00:34:39 --> 00:34:45
What's it to do?
Well, do you see that if a
530
00:34:43 --> 00:34:49
solution gets in that corridor,
an integral curve gets in that
531
00:34:49 --> 00:34:55
corridor, no escape is possible.
It's like a lobster trap.
532
00:34:54 --> 00:35:00
The lobster can walk in.
But it cannot walk out because
533
00:34:58 --> 00:35:04
things are always going in.
How could it escape?
534
00:35:03 --> 00:35:09
Well, it would have to double
back, somehow,
535
00:35:06 --> 00:35:12
and remember,
to escape, it has to be,
536
00:35:10 --> 00:35:16
to escape on the left side,
it must be going horizontally.
537
00:35:17 --> 00:35:23
But, how could it do that
without doubling back first and
538
00:35:20 --> 00:35:26
having the wrong slope?
The slope of everything in this
539
00:35:24 --> 00:35:30
corridor is positive,
and to double back and escape,
540
00:35:28 --> 00:35:34
it would at some point have to
have negative slope.
541
00:35:32 --> 00:35:38
It can't do that.
Well, could it escape on the
542
00:35:35 --> 00:35:41
right-hand side?
No, because at the moment when
543
00:35:39 --> 00:35:45
it wants to cross,
it will have to have a slope
544
00:35:42 --> 00:35:48
less than this line.
But all these spiky guys are
545
00:35:46 --> 00:35:52
pointing; it can't escape that
way either.
546
00:35:50 --> 00:35:56
So, no escape is possible.
It has to continue on,
547
00:35:53 --> 00:35:59
there.
But, more than that is true.
548
00:35:56 --> 00:36:02
So, a solution can't escape.
Once it's in there,
549
00:36:01 --> 00:36:07
it can't escape.
It's like, what do they call
550
00:36:04 --> 00:36:10
those plants,
I forget, pitcher plants.
551
00:36:07 --> 00:36:13
All they hear is they are going
down.
552
00:36:10 --> 00:36:16
So, it looks like that.
And so, the poor little insect
553
00:36:14 --> 00:36:20
falls in.
They could climb up the walls
554
00:36:17 --> 00:36:23
except that all the hairs are
going the wrong direction,
555
00:36:22 --> 00:36:28
and it can't get over them.
Well, let's think of it that
556
00:36:26 --> 00:36:32
way: this poor trap solution.
So, it does what it has to do.
557
00:36:32 --> 00:36:38
Now, there's more to it than
that.
558
00:36:35 --> 00:36:41
Because there are two
principles involved here that
559
00:36:39 --> 00:36:45
you should know,
that help a lot in drawing
560
00:36:43 --> 00:36:49
these pictures.
Principle number one is that
561
00:36:46 --> 00:36:52
two integral curves cannot cross
at an angle.
562
00:36:50 --> 00:36:56
Two integral curves can't
cross, I mean,
563
00:36:53 --> 00:36:59
by crossing at an angle like
that.
564
00:36:56 --> 00:37:02
I'll indicate what I mean by a
picture like that.
565
00:37:02 --> 00:37:08
Now, why not?
This is an important principle.
566
00:37:05 --> 00:37:11
Let's put that up in the white
box.
567
00:37:08 --> 00:37:14
They can't cross because if two
integral curves,
568
00:37:12 --> 00:37:18
are trying to cross,
well, one will look like this.
569
00:37:16 --> 00:37:22
It's an integral curve because
it has this slope.
570
00:37:20 --> 00:37:26
And, the other integral curve
has this slope.
571
00:37:24 --> 00:37:30
And now, they fight with each
other.
572
00:37:27 --> 00:37:33
What is the true slope at that
point?
573
00:37:32 --> 00:37:38
Well, the direction field only
allows you to have one slope.
574
00:37:36 --> 00:37:42
If there's a line element at
that point, it has a definite
575
00:37:40 --> 00:37:46
slope.
And therefore,
576
00:37:41 --> 00:37:47
it cannot have both the slope
and that one.
577
00:37:44 --> 00:37:50
It's as simple as that.
So, the reason is you can't
578
00:37:48 --> 00:37:54
have two slopes.
The direction field doesn't
579
00:37:51 --> 00:37:57
allow it.
Well, that's a big,
580
00:37:53 --> 00:37:59
big help because if I know,
here's an integral curve,
581
00:37:57 --> 00:38:03
and if I know that none of
these other pink integral curves
582
00:38:01 --> 00:38:07
are allowed to cross it,
how else can I do it?
583
00:38:06 --> 00:38:12
Well, they can't escape.
They can't cross.
584
00:38:09 --> 00:38:15
It's sort of clear that they
must get closer and closer to
585
00:38:13 --> 00:38:19
it.
You know, I'd have to work a
586
00:38:16 --> 00:38:22
little to justify that.
But I think that nobody would
587
00:38:20 --> 00:38:26
have any doubt of it who did a
little experimentation.
588
00:38:24 --> 00:38:30
In other words,
all these curves joined that
589
00:38:28 --> 00:38:34
little tube and get closer and
closer to this line,
590
00:38:32 --> 00:38:38
y equals x.
And there, without solving the
591
00:38:37 --> 00:38:43
differential equation,
it's clear that all of these
592
00:38:42 --> 00:38:48
solutions, how do they behave?
As x goes to infinity,
593
00:38:47 --> 00:38:53
they become asymptotic to,
they become closer and closer
594
00:38:52 --> 00:38:58
to the solution,
x.
595
00:38:54 --> 00:39:00
Is x a solution?
Yeah, because y equals x is an
596
00:38:58 --> 00:39:04
integral curve.
Is x a solution?
597
00:39:02 --> 00:39:08
Yeah, because if I plug in y
equals x, I get what?
598
00:39:07 --> 00:39:13
On the right-hand side,
I get one.
599
00:39:10 --> 00:39:16
And on the left-hand side,
I get one.
600
00:39:14 --> 00:39:20
One equals one.
So, this is a solution.
601
00:39:18 --> 00:39:24
Let's indicate that it's a
solution.
602
00:39:21 --> 00:39:27
So, analytically,
we've discovered an analytic
603
00:39:26 --> 00:39:32
solution to the differential
equation, namely,
604
00:39:31 --> 00:39:37
Y equals X, just by this
geometric process.
605
00:39:37 --> 00:39:43
Now, there's one more principle
like that, which is less
606
00:39:41 --> 00:39:47
obvious.
But you do have to know it.
607
00:39:44 --> 00:39:50
So, you are not allowed to
cross.
608
00:39:46 --> 00:39:52
That's clear.
But it's much,
609
00:39:49 --> 00:39:55
much, much, much,
much less obvious that two
610
00:39:52 --> 00:39:58
integral curves cannot touch.
That is, they cannot even be
611
00:39:57 --> 00:40:03
tangent.
Two integral curves cannot be
612
00:40:00 --> 00:40:06
tangent.
613
00:40:02 --> 00:40:08
614
615
616
00:40:10 --> 00:40:16
I'll indicate that by the word
touch, which is what a lot of
617
00:40:19 --> 00:40:25
people say.
In other words,
618
00:40:23 --> 00:40:29
if this is illegal,
so is this.
619
00:40:28 --> 00:40:34
It can't happen.
You know, without that,
620
00:40:33 --> 00:40:39
for example,
it might be,
621
00:40:35 --> 00:40:41
I might feel that there would
be nothing in this to prevent
622
00:40:39 --> 00:40:45
those curves from joining.
Why couldn't these pink curves
623
00:40:43 --> 00:40:49
join the line,
y equals x?
624
00:40:45 --> 00:40:51
You know, it's a solution.
They just pitch a ride,
625
00:40:49 --> 00:40:55
as it were.
The answer is they cannot do
626
00:40:52 --> 00:40:58
that because they have to just
get asymptotic to it,
627
00:40:55 --> 00:41:01
ever, ever closer.
They can't join y equals x
628
00:40:59 --> 00:41:05
because at the point where they
join, you have that situation.
629
00:41:05 --> 00:41:11
Now, why can't you to have
this?
630
00:41:09 --> 00:41:15
That's much more sophisticated
than this, and the reason is
631
00:41:17 --> 00:41:23
because of something called the
Existence and Uniqueness
632
00:41:24 --> 00:41:30
Theorem, which says that there
is through a point,
633
00:41:31 --> 00:41:37
x zero y zero,
that y prime equals f of
634
00:41:38 --> 00:41:44
(x, y) has only one,
635
00:41:43 --> 00:41:49
and only one solution.
One has one solution.
636
00:41:49 --> 00:41:55
In mathematics speak,
that means at least one
637
00:41:53 --> 00:41:59
solution.
It doesn't mean it has just one
638
00:41:56 --> 00:42:02
solution.
That's mathematical convention.
639
00:41:59 --> 00:42:05
It has one solution,
at least one solution.
640
00:42:02 --> 00:42:08
But, the killer is,
only one solution.
641
00:42:06 --> 00:42:12
That's what you have to say in
mathematics if you want just
642
00:42:10 --> 00:42:16
one, one, and only one solution
through the point
643
00:42:15 --> 00:42:21
x zero y zero.
So, the fact that it has one,
644
00:42:18 --> 00:42:24
that is the existence part.
The fact that it has only one
645
00:42:23 --> 00:42:29
is the uniqueness part of the
theorem.
646
00:42:26 --> 00:42:32
Now, like all good mathematical
theorems, this one does have
647
00:42:31 --> 00:42:37
hypotheses.
So, this is not going to be a
648
00:42:35 --> 00:42:41
course, I warn you,
those of you who are
649
00:42:39 --> 00:42:45
theoretically inclined,
very rich in hypotheses.
650
00:42:44 --> 00:42:50
But, hypotheses for those one
or that f of (x,
651
00:42:48 --> 00:42:54
y) should be a
continuous function.
652
00:42:52 --> 00:42:58
Now, like polynomial,
signs, should be continuous
653
00:42:57 --> 00:43:03
near, in the vicinity of that
point.
654
00:43:02 --> 00:43:08
That guarantees existence,
and what guarantees uniqueness
655
00:43:08 --> 00:43:14
is the hypothesis that you would
not guess by yourself.
656
00:43:14 --> 00:43:20
Neither would I.
What guarantees the uniqueness
657
00:43:19 --> 00:43:25
is that also,
it's partial derivative with
658
00:43:24 --> 00:43:30
respect to y should be
continuous, should be continuous
659
00:43:30 --> 00:43:36
near x zero y zero.
660
00:43:35 --> 00:43:41
Well, I have to make a
decision.
661
00:43:38 --> 00:43:44
I don't have time to talk about
Euler's method.
662
00:43:43 --> 00:43:49
I'll refer you to the,
there's one page of notes,
663
00:43:49 --> 00:43:55
and I couldn't do any more than
just repeat what's on those
664
00:43:55 --> 00:44:01
notes.
So, I'll trust you to read
665
00:43:59 --> 00:44:05
that.
And instead,
666
00:44:02 --> 00:44:08
let me give you an example
which will solidify these things
667
00:44:09 --> 00:44:15
in your mind a little bit.
I think that's a better course.
668
00:44:17 --> 00:44:23
The example is not in your
notes, and therefore,
669
00:44:22 --> 00:44:28
remember, you heard it here
first.
670
00:44:27 --> 00:44:33
Okay, so what's the example?
So, there is that differential
671
00:44:34 --> 00:44:40
equation.
Now, let's just solve it by
672
00:44:38 --> 00:44:44
separating variables.
Can you do it in your head?
673
00:44:42 --> 00:44:48
dy over dx, put all the y's on
the left.
674
00:44:44 --> 00:44:50
It will look like dy over one
minus y.
675
00:44:48 --> 00:44:54
Put all the dx's on the left.
So, the dx here goes on the
676
00:44:52 --> 00:44:58
right, rather.
That will be dx.
677
00:44:54 --> 00:45:00
And then, the x goes down into
the denominator.
678
00:44:57 --> 00:45:03
So now, it looks like that.
And, if I integrate both sides,
679
00:45:03 --> 00:45:09
I get the log of one minus y,
I guess, maybe with a,
680
00:45:08 --> 00:45:14
I never bothered with that,
but you can.
681
00:45:12 --> 00:45:18
It should be absolute values.
All right, put an absolute
682
00:45:17 --> 00:45:23
value, plus a constant.
And now, if I exponentiate both
683
00:45:23 --> 00:45:29
sides, the constant is positive.
So, this is going to look like
684
00:45:29 --> 00:45:35
y.
One minus y equals x
685
00:45:33 --> 00:45:39
And, the constant will be e to
686
00:45:36 --> 00:45:42
the C1.
And, I'll just make that a new
687
00:45:39 --> 00:45:45
constant, Cx.
And now, by letting C be
688
00:45:42 --> 00:45:48
negative, that's why you can get
rid of the absolute values,
689
00:45:45 --> 00:45:51
if you allow C to have negative
values as well as positive
690
00:45:49 --> 00:45:55
values.
Let's write this in a more
691
00:45:51 --> 00:45:57
human form.
So, y is equal to one minus Cx.
692
00:45:53 --> 00:45:59
Good, all right,
693
00:45:55 --> 00:46:01
let's just plot those.
So, these are the solutions.
694
00:46:00 --> 00:46:06
It's a pretty easy equation,
pretty easy solution method,
695
00:46:05 --> 00:46:11
just separation of variables.
What do they look like?
696
00:46:11 --> 00:46:17
Well, these are all lines whose
intercept is at one.
697
00:46:16 --> 00:46:22
And, they have any slope
whatsoever.
698
00:46:19 --> 00:46:25
So, these are the lines that
look like that.
699
00:46:24 --> 00:46:30
Okay, now let me ask,
existence and uniqueness.
700
00:46:29 --> 00:46:35
Existence: through which points
of the plane does the solution
701
00:46:35 --> 00:46:41
go?
Answer: through every point of
702
00:46:39 --> 00:46:45
the plane, through any point
here, I can find one and only
703
00:46:44 --> 00:46:50
one of those lines,
except for these stupid guys
704
00:46:48 --> 00:46:54
here on the stalk of the flower.
Here, for each of these points,
705
00:46:53 --> 00:46:59
there is no existence.
There is no solution to this
706
00:46:57 --> 00:47:03
differential equation,
which goes through any of these
707
00:47:02 --> 00:47:08
wiggly points on the y-axis,
with one exception.
708
00:47:07 --> 00:47:13
This point is oversupplied.
At this point,
709
00:47:10 --> 00:47:16
it's not existence that fails.
It's uniqueness that fails:
710
00:47:14 --> 00:47:20
no uniqueness.
There are lots of things which
711
00:47:18 --> 00:47:24
go through here.
Now, is that a violation of the
712
00:47:21 --> 00:47:27
existence and uniqueness
theorem?
713
00:47:24 --> 00:47:30
It cannot be a violation
because the theorem has no
714
00:47:28 --> 00:47:34
exceptions.
Otherwise, it wouldn't be a
715
00:47:31 --> 00:47:37
theorem.
So, let's take a look.
716
00:47:34 --> 00:47:40
What's wrong?
We thought we solved it modulo,
717
00:47:37 --> 00:47:43
putting the absolute value
signs on the log.
718
00:47:40 --> 00:47:46
What's wrong?
The answer: what's wrong is to
719
00:47:43 --> 00:47:49
use the theorem you must write
the differential equation in
720
00:47:48 --> 00:47:54
standard form,
in the green form I gave you.
721
00:47:51 --> 00:47:57
Let's write the differential
equation the way we were
722
00:47:54 --> 00:48:00
supposed to.
It says dy / dx equals one
723
00:47:57 --> 00:48:03
minus y divided by x.
724
00:48:02 --> 00:48:08
And now, I see,
the right-hand side is not
725
00:48:05 --> 00:48:11
continuous, in fact,
not even defined when x equals
726
00:48:09 --> 00:48:15
zero, when along the y-axis.
And therefore,
727
00:48:12 --> 00:48:18
the existence and uniqueness is
not guaranteed along the line,
728
00:48:16 --> 00:48:22
x equals zero of the y-axis.
And, in fact,
729
00:48:20 --> 00:48:26
we see that it failed.
Now, as a practical matter,
730
00:48:23 --> 00:48:29
it's the way existence and
uniqueness fails in all ordinary
731
00:48:28 --> 00:48:34
life work with differential
equations is not through
732
00:48:32 --> 00:48:38
sophisticated examples that
mathematicians can construct.
733
00:48:38 --> 00:48:44
But normally,
because f of (x,
734
00:48:40 --> 00:48:46
y) will fail to be
defined somewhere,
735
00:48:43 --> 00:48:49
and those will be the bad
points.
736
00:48:46 --> 00:48:52
Thanks.
| 677.169 | 1 |
unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. Written for trainers and participants of contests of all levels up to the highest level, this will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a "problem of the week", thus bringing a creative atmosphere into the classrooms. Equally, this is a must-have for individuals interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. In addition to being a valuable resource of mathematical problems and solution strategies, this is the most complete training book on the market.
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Each chapter is full of sample exercises and end by around a 100 problems to solve making the total number of problems in the book to 1300. The problems are selected to illustrate techniques in difficult and non-routine problem solving using problems from past IMO, Tournament of the Towns non Calculus Putnam problems and National competitions from many countries. Happy Problem Solving!
A must for participants in math contests and their trainers, and a real treasure for all math lovers and problem-solving fans. The author focuses on the main ideas, techniques and strategies needed to solve the kind of problems found at "elementary" math competitions, up to the IMO level. With more than 1300 problems and examples, it is also an excellent source for teachers in search of interesting, non-routine problems to challenge their students, stimulate their creativity or even to motivate the study of some subjects. My only concern is that, at the sight of such abundance of material, some students might be overwhelmed or discouraged. Ideally, a qualified teacher should select the problems and assign them in adequate doses to the math strength of their students.
Firstly, this book is probably not well suited for the beginner. It is definitely a comprehensive presentation of elementary and ingenious problem solving methods. Techniques such as Pigeonhole(box principle), invariants, Plane and transformational geometries, coloring proof, number theory, enumarative combinatorics, and quite a few more are presented with many(MANY!) examples and problems. This is definitely the most complete book on mathematical technique that I have seen to date. This book is magnificent. Highly Recommend
Problem-Solving Strategies is a great book for anyone interested in Mathematical Contests. It is not however a book for starters but rather a book for university students and high-school contest participants.
I am a student at a college in New England and I bought this book in high school. The book claims no knowledge of higher mathematics is necessary to complete the problems but that claim is simply untrue. You need higher mathematics to solve some of these problems. The author says anyone with a standard high school mathematics education can solve these problems. I strongly disagree. They are similar to high school problems but they are in much much greater depth.
Wow! The problems in the book are extremely difficult but not impossible. I would recommend only buying this book if you have access to mathematics professor(s) who can help you understand these problems for the Putnam Competition or you have a committed group of friends who like math. It takes me hours to solves these problems but I love it!
This is a rather complete book for students and coaches who are preparing for mathematics competitions such as the IMO. It is loaded with problem-solving strategies for a large variety of mathematics problems ranging from combinatorics, number theory, sequences, polynomials and geometry to game theory, a topic which is not normally included in mathematics competitions. There are tons of problems (around 1300) to hone your mathematical skills. Another very good book in a similar category is the IMO Compendium. The books by Titu Andreescu are also excellent for mathematics competitions.
My own field is computer science but love reading mathematics for its purity and occasional aha moments. This one I bought for my son at his request and I must admit that I was the first one to peruse through it. In my cursory reading I did enjoy going through each of the chapters. The author has a good command over his teaching style and he peppers it with thrilling anecdotes. It should definitely be seen on the desk of any serious mathematics student (at least at high school level). But I did also find some errors randomly and I wanted to exchange notes with others but sadly I could not find any site. Even springer does not have any errata listed for this book. Now that the book is out of the press, the publishers need to show some compassion for the buyers and issue an errata soon.
This is one book that I would like to keep with me all the time even after all my interest in Maths. is gone. For, it gives me food for thought and does not allow my mind to be dried of thoughts. This is a most fascinating book since the author, even when he gives hints at the solution, he does it in such an effective manner, that you have to really pursue a very thrilling chase of the solution. The challenge and thrill make it a pleasurable experience. I am sure this will be so for both the learner and the learned. Every mathematician should possess this book.
| 677.169 | 1 |
Product Description "investigations."
The "Tests and Worksheets" book provides a "facts practice test" for each lesson as well as a test covering 5-10 lessons. Investigations are also included.
The solutions manual provides answers for all problems in the lesson (including Warm-up, lesson practice, and mixed practice exercises), as well as solutions for the supplemental practice found in the back of the student text, and the facts practice tests, activity sheets, and tests in the separate tests & worksheets book.
Product Reviews
Saxon Math 76 Home Study Kit Fourth Edition
4.9
5
44
44
This is the third Saxon book we have used in homeschooling over the past 2 and a half years. There are no fancy illustrations, and I know that the lessons are delivered in a predictable format, but we really like the program. Our son is sharp, and he can still benefit from the spiral approach of Saxon. The fact that previous lessons continue to be integrated into current lessons really helps solidify our 12 year old's comprehension. Many days we assign two lessons, and circle which problems we want him to complete on the mixed practice. The time tests are very beneficial too. In preparation for this year, his sixth grade year, I reviewed, and even tested him in Teaching Textbooks and Singapore. He was above the recommended level in TT, and I just wanted more problems than Singapore had in its text. We really like Saxon, and our son begins every day with math, whether I am present to get him started or not. That says a lot.
February 4, 2014
Excellent!
Love this curriculum! I discovered in when I was working in Juvenile Detention Center and was amazed at the results I saw with teens that had significant learning disabilities, hated school, and of course math! After couple of weeks on this curriculum, most were making A's and the confidence they were getting, reflected on their behavior. Now I use it on my kids and they love it, although my daughter is disappointed that I am making her start at a lower level, but public school math left so many holes, that we are doing alot of back work before we can successfully move forward. Thank you Saxon!
November 12, 2013
Love It!!
There is no better Math text than the Saxon series. I introduced it in the 1980's in the Christian high school were our two older daughters were in my class. I have never seen anything to rival the approach Saxon takes. I have a Masters Degree in Teaching Math and enough years of teaching all levels and courses from K-12 and tutoring teens and adults through college and graduate school. The Saxon approach is where the rubber meets the road for giving a student the best foundation and tools for success in any field of study and just plain old everyday life. You will not find a Saxon student who grows up to hate Math. They will be the parents who love to help their kids with homework problems, too.
September 12, 2013
Do you want your child overly prepared?
The Saxon Curriculum prepares your child years above their public school counterparts. After Algebra II in this program for my daughter I put her in public school for Advanced Math. She was so far ahead of where they were and where they were going that I pulled her out and ordered Saxon Advanced Math for her to complete at home. The curriculum prepares our children far ahead of the same curriculum taught elsewhere. You will not be disappointed. I will never stray again. My daughter is now in Calculus in the College and my son will be following in her footsteps because of Saxon.
April 24, 2013
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I'm studying algorithms complexities by myself (my university didn't it to me) and I'd love if someone could help me in finding good resources to learn fundamental algorithms complexities proofing. There are lots of non-mathematical resources about big-Oh notation but I'd like something more mathematical and rigorous (although understandable) on the topic
1 Answer
As a double major in maths and CS I liked Introduction to Algorithms by cormen et al., when I was taught about algorithms for the first time.
This book tries to be complete, therefore it was sometimes too mathematical for the general CS audience so parts were skipped in my algorithm course(proof of the Master Theorem e.g.).
Besides the rigor rarely seen in CS textbooks, there is a vast amount of algorithms which are explained. Because of this, it is the CS book I own I pick up the most, mostly for some algorithm I want to review.
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Video Description: Herb Gross introduces Calculus Revisited II — Functions of Several Variables — and discusses the over-arching theme "The Game of Mathematics". A game has objectives, rules, and definitions as well as strategies (logical plans) for meeting the objectives of the game. A mathematical structure is identical with the objective being to model real-world experience
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Why MAPLE?
Mathematics is more than just arithmetic. It is about symbolic relationships between numbers and more complicated mathematical quantities based on numbers. Computer algebra systems (CAS) were designed as a tool for handling symbolic mathematical quantities and relations exactly, as well as in representing their numerical and graphical aspects, with mathematical word processing (verbal) later added as a bonus to weave all three of these aspects of mathematics (algebraic, numerical, graphical) together into a powerful platform for effective mathematical calculation and communication, backed up by an enormous amount of mathematical knowledge available to the intelligent user. Without understanding the mathematical concepts underlying its rich command structure, it is a useless tool—it does not substitute for learning the mathematical foundations, although it does free us from many routine and often pointless calculations, freeing us to give our attention to charting a path towards the solution of the real mathematical problems which confront us. It also gives us the opportunity to visualize many crucial ideas that can make dry mathematical notation come alive.
A computer algebra system (as opposed to other kinds of mathematical software) is the clear choice as the primary technology tool for aiding learning in a science/engineering based calculus, differential equations and linear algebra sequence, as well as in higher level mathematics courses (although more specialized tools are appropriate for advanced statistical applications). This is not to say that other choices of mathematical software are not also important in applications of this mathematics to science and engineering. Indeed Excel, MathCAD, and MatLab (all in use here at Villanova) all have important roles to play in various technical disciplines. Our choice of CAS is Maple, perhaps the most widely used CAS in college education (together with Mathematica, also in use here at Villanova).
Goals in the Educational Process
Our goal in using Maple in this course sequence (1500, 1505, 2500, 2705) is not to teach Maple for its own sake, but to use it as a tool in aiding your learning. Since some familiarity with a CAS is an important part of your background as a student in science and engineering in the 21st century, what you do learn about using Maple can and should prove useful even after you complete these courses.
In order to be able to use Maple as a tool to aid learning, it is important to get familiar with the most useful aspects of its worksheet interface which a little Windows environment savvy and exploration easily accomplishes. Any mathematical activity that is not able to be addressed by intuitively building standard mathematical expressions and referring to a few help tips should be guided with a template from your instructor or a textbook technology supplement. By asking your instructor when necessary, and referring to Maple help and MLRC help personnel when necessary, whatever difficulty you have in using Maple can be overcome. There is no need to be frustrated. Work with a partner, and if you get stuck, get help but do not waste time.
Fortunately in Standard Maple a reliable extremely useful set of palettes and context dependent right-click menus have been introduced ("clickable calculus") which enable a new user who knows no Maple syntax to do almost all the calculations needed for the calculus, differential equations and linear algebra sequence courses in which it is a required tool. Within Standard Maple, the Tools Menu, Tutors submenu combines these features into an even more powerful popup applet window for many basic concepts that allows parameters for the activity to be set in the window, delivering a final result to the worksheet, but allowing mouse selection, Control C copying, and Control V pasting of extra window content into text regions of the worksheet as well. Feedback from Maple helps unblock a student who does not see how to choose the next step in a multi-step Tutor activity. Similarly the Tools Menu, Tasks, Browse option allows you to browse the Calculus list of typical activities that are provided with worksheet examples that may be easily inserted into your worksheet where you need them.
While outside projects using Maple can enhance your ability to use Maple and show you how it can help you solve much more interesting and realistic problems, it is important to use Maple in selected homework problems from your textbook on a regular basis so that it can help you in learning the concepts. In order to be effective, these problems should be chosen carefully keeping in mind the goal of enhancing your learning of the concepts, although it is also useful to do some routine problems to get familiar with using certain commands to substitute for the "back of the book" odd answer key as a check on your hand calculations.
For general tips on using Standard Maple and example files for the MAT1500-1505-2500-2705 sequence see
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038790459X
9780387904597
Elementary Analysis:Designed for students having no previous experience with rigorous proofs, this text can be used immediately after standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, as well as for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied, while many abstract ideas, such as metric spaces and ordered systems, are avoided completely. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics, and optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals.
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Rent Elementary Analysis 12th edition today, or search our site for Kenneth A. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Springer.
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their maths studies. 1 TYPES OF BEAMS A beam is a structure, which is loaded transversely (sideways). ... programmes for solving beamproblems. The Archon Engineering web site has many such programmes. WORKED EXAMPLE No.1
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Mental Maths & Problem Solving Rebeca Muñoz San Millán ... Listen to the problems and complete these sets according to the operations: Set 1 Set 4 Set 2 Set 3 Set 4 ... 2.-Add standard weights to the other pan until the beam is balance
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Maths has applications to many problems that are vital to human health and happiness. In this article we are going to describe how the mathematics of tomography has ... If a light beam is shone through several bottles, then this absorption adds up.
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Use geometric reasoning to solve problems A ... 45° Ground Narrow Beam Ladder Ladder It is built on sloping ground. The beam, AB, is parallel to the ground. The support posts, AE and BD, are vertical. The support post, AE, makes an angle of 80º with the ground.
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radiographers sometimes use X-ray beam filters and need to be able to quantify the impact of filtration on the intensity and quality of the resulting images, as well as on patient exposure. Once the images are processed, radiographers' next
A school website that contains lots of maths games and activities ... Solve problems involving addition, subtraction, multiplication or division in contexts of numbers, measures or pounds and pence
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Despite the introduction of the National Curriculum, problems appear to remain ... New Curriculum maths, BEAM and SPMG Primary School 4 Cambridge maths, Mental maths and Ginn Primary school survey all schools use some scheme, Cambridge maths used by
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Two distinct problems are considered: the first is where the stress is assumed continuous across the boundary ... There, the solution is simplified by assumingthat the turning angle α throughwhich the beam is bent is specified,
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1 Star Problems ADD TO ONE-THOUSAND Problem There are exactly three different pairs of positive integers that add to make six. 1 + 5 = 6 ... If a beam of light is fired through the top left corner of a 2 by 50 rectangle, which corner will it emerge from?
All the required terms may be computed beforehand (by using a symbolic maths program) and then stored and encoded in the program developed. ... The first of these problems is that of a beam on an elastic foundation subjected to a static load. Then, ...
mathematics, problems drawn from other subjects studied in the ... problem relating to beam-columns. The problem requires students to derive a set of finite difference equations for a ... Maths for Engineering and Science
3.3 Problems ... Before introducing more complicating maths, an example of the above variation in ... For the beam of Example 16, show that the stiffness of the spring support that optimizes the bending moments is 89.5EIl3, i.e. makes the sagging and
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Limits, Limits Everywhere: The Tools of Mathematical Analysis
by David Applebaum Publisher Comments
A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an... (read more)
Indesign Cs5 for Dummies
by Galen Gruman Publisher Comments
Get up to speed on InDesign CS5 and create great publicationsfor print or the Web How do you design? Free-form, creating one-of-a-kind pieces? Or would you love a highly formatted template that you could modify as needed for regular periodicals? InDesign... (read more)
Network+ Guide to Networks
by Tamara Dean Publisher Comments
Network+ Guide to Networks, Fourth Edition is designed to prepare users for CompTIA's newly-revised 2005 Network+ certification exam and will also offer mapping features to the exam objectives....Algebra II for Dummies (For Dummies)
by Mary Jane Sterling Publisher Comments
Besides being an important area of math for everyday use, algebra is a passport to studying subjects like calculus, trigonometry, number theory, and geometry, just to name a few. To understand algebra is to possess the power to grow your skills andFundamentals of Acoustics
by Lawrence E Kinsler Publisher Comments
A clear treatment of the fundamental principles underlying the generation, transmission, and reception of acoustic waves and their application to numerous fields. Analyzes the various types of vibration of solid bodies and the propagation of sound waves... (read more)
The History of the Calculus and Its Conceptual Development
by Carl B Boyer Publisher Comments
This book, for the first time, provides laymen and mathematicians alike with a detailed picture of the historical development of one of the most momentous achievements of the human intellect ― the calculus. It describes with accuracy and... (read more)
Lambda-Matrices and Vibrating Systems
by Peter Lancaster Publisher Comments
Features aspects and solutions of problems of linear vibrating systems with a finite number of degrees of freedom. Starts with development of necessary tools in matrix theory, followed by numerical procedures for relevant matrix formulations and... (read more)
The Grand Design
by Stephen Hawking and Leonard Mlodinow Publisher Comments
#1 New York Times Bestseller When and how did the universe begin? Why are we here? What is the nature of reality? Is the apparent "grand design" of our universe evidence of a benevolent creator who set things in motion — or does science
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Calculus and Its Applications has, for years, been a best-selling text for one simple reason: it anticipates, then meets the needs of today's applied calculus student. Knowing that calculus is a course in which students typically struggle--both with algebra skills and visualizing new calculus concepts--Bittinger and Ellenbogen speak to students in a way they understand, taking great pains to provide clear and careful explanations.&Since most&students taking this course will go on to careers in the business world, &large quantities of real data, especially as they apply to&business, are included as well.
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Peh's Teaching Page
My teaching philosophy is based on my unequivocal belief that
education is a never-ending two-way process .
With the students' willingness and commitment to learn, I will
do my best to guide my students in developing the following
abilities, namely, to think and analyze
mathematical problems,
to communicate with people in other fields,
to be self-sufficient, and to not be afraid to be innovative even
if it means making mistakes sometimes.
In return , I expect myself to learn as much from my students as they
do from me.
Based on my many years of teaching experience,
I found that that there are ideal conditions under which
students will learn effectively, namely, when:
the students are actively participating in class,
the students are willing to put forth efforts into the course while
they are outside the classroom,
the course material comes across with pertinence,
I, the instructor, can motivate and communicate
the subject matter by using various ways that will
facilitate participation by the students and
I, the instructor, am able to
show the pertinence
of the subject
In addition, within the subject matter of mathematics,
I am convinced that different people have different learning styles
with which they grasp these concepts.
Thus, without losing any rigor on the subject matter,
whenever I teach I make sure that I include these
components of these styles so as to maximize the learning
capabilities of my students on the subject matter.
In other words, the students must:
see ideas/concepts in written forms
hear ideas/concepts spoken by someone else
write the ideas/concepts down themselves
have the chance to think about the ideas/concepts themselves
speak about or explain the ideas/concepts to others
practice using the ideas/concepts in various different scenarios
In the lower level math courses,
since more students have a less
solid background in theoretical mathematics, the strategies used are
lectures, the first few minutes of which are usually
question/answer discussions, and daily assignments.
The reason for the daily assignments and weekly pop quizzes is so that
students
have enough practice and maintain a up-to-date schedule with the
course materials so as to feel more
comfortable and confident in the subject matter.
In addition I also give a few more elaborate problem sets
where students have to make use of technology to obtain
numerical solutions.
Ocassionally, I will ask them to work a few challenging problems in
groups (in class or outside of class) and ask them to write the
solutions up, formally, and to present them in class.
During my lectures, I always make the process an interactive
form instead of just myself talking and the students writing.
In other words, I try to lead my students into new concepts instead
of just telling them.
I let my students know from the very beginning that I strongly
welcome any questions pertaining to the subject, and that I do not
consider any question
about the material as stupid .
Since assignments in my upper level courses are more elaborate,
I give them on a weekly basis.
Course projects (groups and individual) are mandatory in all
upper level applied math courses I teach. The purpose of such
course projects is for students to be aware of how and where concepts
of the courses are applied beyond the walls of academia.
I also require students to turn in
written reports and to give oral presentations about their
course projects.
Brief and Precise Description of my teaching interests
In a nutshell, I am a die-hard, self-avowed and
practising applied mathematician
who is interested in teaching a variety of undergraduate
math courses, including the following:
Calculus and its applications
Discrete & Combinatorial Mathematics
Operations Research
Linear Algebra and applications
Mathematical Programming
Network Modeling and Optimization
Mathematical Modeling
Graph Theory and Applications
Matroid Theory
In general, my areas of teaching interests include, but are not
limited
to, diverse areas in the field of operations research and its applications,
namely, most of the aforementioned areas.
For more information on these topics, try to either take or audit
the following scintillating mathematics courses, namely,
Discrete & Combinatorial Math (Math 3411, or
Math 1760 & 3370 under quarters), and
Operations Research
(Math 3401, or
Math 3270 under quarters).
These areas of mathematics have a plethora of
real-world applications and applications in other
areas of theoretical mathematics.
If you would like to learn more,
please feel free to talk to me or take or audit the courses I teach.
Homepage of MA3370, a.k.a.
Combinatorial Mathematics, . F.Y.I., this is a course with a service-learning component, i.e. a way for UMM students and faculty to reach out and touch the great City of Morris ;-)
nal
Homepage of MA5900, a.k.a.
Discrete Mathematics & Graph Theory.
( This is a workshop course for secondary math teachers in conjuction with the (ME)^3 Project, sponsored by the Minnesota Board of Higher Education ).
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Take the guesswork out of high school math instruction!
Quickly and reliably uncover common math misconceptions in Grades 9-12 with these convenient and easy-to-implement diagnostic tools! Bestselling authors Cheryl Rose Tobey and Carolyn B. Arline provide 25 new assessment probes that pinpoint subconcepts within the Common Core State StandardsGet your best grade with My Revision Notes: AQA A2 Law: Criminal Law Units 3A and 4A and Concepts of Law Unit 4C. Unlock your full potential with this revision guide which focuses on the key content and skills you need to know for the Criminal Law and Concepts of Law papers for AQA A2 Law. With My Revision Notes: AQA A2 Law: Criminal and Concepts... more...
Get your best grade with My Revision Notes: AQA AS Law. Unlock your full potential with this revision guide which focuses on the key content and skills you need to know for AQA AS Law. With My Revision Notes for AQA AS Law you can:. - Take control of your revision: plan and focus on the areas you need to revise with content summaries and exam tips.... more...
Unlock your full potential with these revision guides which focus on the key content and skills you need to know. With My Revision Notes for AQA GCSE Religious Studies Religion and Life Issues and Religion and Morality you can:. - Take control of your revision: plan and focus on the areas you need to revise with content summaries and commentary from... more...
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Mathematics A Discrete Introduction
9780534356385
ISBN:
0534356389
Pub Date: 2000 Publisher: Brooks/Cole
Summary: This book is an introduction to mathematics--in particular, it is an introduction to discrete mathematics. There are two primary goals for this book: students will learn to reading and writing proofs, and students will learn the fundamental concepts of discrete mathematics.
Scheinerman, Edward A. is the author of Mathematics A Discrete Introduction, published 2000 under ISBN 9780534356385 and 0534356389. Twe...nty three Mathematics A Discrete Introduction textbooks are available for sale on ValoreBooks.com, eighteen used from the cheapest price of $1.00, or buy new starting at $26.19
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Loren: in a group theory course intended primarily for juniors and seniors planning to become high school mathematics teachers I find that links with geometry, especially symmetries of frieze and wallpaper patterns as well as platonic solids, illustrate the ideas in a way that students can appreciate readily. A good text for this is
"Groups and Symmetry" by David W. Farmer
published recently by the AMS. This can be combined with a Dynamic Geometry software program such as Geometers SketchPad to give students opportunities for exploring group theory concepts.
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Open Textbook
free algebra textbook from Boundless Learning is based off openly available educational resources such as "government resources, open educational repositories, and other openly licensed websites." The textbook contains 8 chapters such as The Building Blocks of Algebra, Graphs, Functions, and Models, and Conic Sections. The textbook can be browsed on this page or downloaded as a pdf. Students can register for a free Boundless account to access a search engine and other study tools to efficiently find specific topics and master the content.Mon, 3 Mar 2014 16:01:31Mathematics Assessment: A Video Library
educators will find much to enjoy on this clutch of video programs created to help illustrate the link between instruction and assessment. Created by WGBH Boston, this 11-part series includes case studies that involve animals in Yellowstone, fractions, and the geometry of a Ferris Wheel. The last program, Beyond Testing, is most efficacious as it offers background information on assessment issues along with some questions for further discussion. Additionally, the site contains links to related resources and other video series from WGBH and the Annenberg Learner Foundation.Tue, 14 Jan 2014 11:05:26 -0600GeoGebra
you're a mathematics educator, you'll find the GeoGebra site to be a perfect addition to your stable of online resources. On the top of the homepage, visitors can look through sections that include About, Download, Community, and Materials. Most visitors will want to download the GeoGebra application as it is the primary way to utilize the 4,400 learning activities offered on the site. After completing the download of this free, interactive geometry, algebra, statistics, and calculus software, click on over to the Featured Materials area. Here visitors can look over the newest materials, check out the Best Worksheets, and wander through the popular tags. Mathematics educators have contributed many of the items here, and a small sample includes "The 11 Patterns of the Cube" and "Introduction to Linear Equations." Finally, visitors can also use the Community area to ask questions and share resources.Thu, 3 Oct 2013 10:27:37 -0500National Council of Teachers of Mathematics
the best way to teach young people about geometry? Or general data analysis? The National Council of Teachers of of Mathematics (NCTM) has a few ideas on the subject and they have brought them together in the Core Math Tools Suite. This downloadable suite of interactive software tools for algebra, geometry, statistics, and related topics can be used in a range of educational settings. The General Purpose Tools area contains five tools that require strategies and skills that are highly applicable to a range of analytical skills. The site also contains How-To Pages, along with Advanced Apps such as the "Ranked Choice Voting" app, which can be used to determine voting outcomes based on various ranked choice voting methods.Thu, 12 Sep 2013 11:52:00 -0500Modeling Exponential Growth
lesson plan from ATEEC will help students learn the concept of exponential growth, as well as interpret and apply exponential relationships. It is intended for environmental science or biology classes. The activity sheet includes a list of materials and resources needed as well as a description of the lesson. This resource is free to download from ATEEC; users must create a free login to access.Tue, 9 Jul 2013 13:44:54 -0500Webmath
from Discovery Education, provides help for mathematics students. Categories include general mathematics, K-8 math, algebra, geometry, trigonometry and calculus. The site covers everything you need to know, whether you need help with a specific topic or are looking to brush up on some math skills.Mon, 10 Jun 2013 13:23:38 -0500West Texas A&M University Virtual Math Lab
page from West Texas A&M University provides help for students in college algebra, intermediate algebra, beginning algebra, math for the sciences, and GRE mathematics preparation. Each area contains a number of individual tutorials that increase in difficulty. The GRE prep section includes two practice tests. This would be a wonderful resource for people looking to brush up on their math skills or students returning to college after some time away from math classes.Wed, 5 Jun 2013 13:13:34 -0500National Security Agency: High School Concept Development Units
National Security Agency (NSA) has worked to craft these educational materials they are calling "concept development units" (CDUs). The units are divided into 11 sections, including Algebra, Calculus, and Data Analysis. Clicking on each of these sections will bring up a complete list of all the CDUs currently available. Each list offers a paragraph-long description of each activity, along with an indication of the appropriate grade level for each activity. Some of the activities include "Understanding Proportions and Scale Drawings," "Scatter Brained," "Fashion Sense and Dollar Wise" and "Squares in the Light." These are all terrific resources for educators, and the site also contains links to information about the Math and Related Sciences Camp (MARS) sponsored by the National Security Agency and links to other educational centers.Fri, 15 Feb 2013 10:50:45 -0600Mathematics for Photonics Education
has developed program planning and course materials to support education and training for future and current photonics technicians. Classroom materials include Scientific Notification, Unit Conversion, Introductory Algebra, Introductory Geometry, Introductory Trigonometry, Exponents, Logarithms, and Graphing. Visitors can request evaluation copies of any of these materials or purchase them via the contact information given.Mon, 22 Oct 2012 10:56:35 -0500Get the Math
does math get used in the "real world?" The short answer is that it is used to create hip-hop music, in fashion design, and through a number of other endeavors. This interactive website combines video and web interactive to help young people develop algebraic thinking skills for solving real-world problems. The series is funded by The Moody's Foundation, along with assistance from WNET and American Public Television. The sections of the site include The Challenges, Video, and Teachers. In The Challenges area, users will find video segments profiling the various young professionals who use math in their work, along with interactive tools to help students solve the challenges they are presented with. Moving on, the Teachers area includes resources for teachers, such as a training video showing how to use project materials in the classroom, along with student handouts. Visitors shouldn't miss the Basketball challenge, featuring NBA player Elton Brand talking about the problems presented by free throw shooting.Wed, 27 Jun 2012 10:54:56 -0500Bates College Online Resources for Calculus and Linear Algebra
College in Maine has worked diligently to bring together this set of mathematical resources to the public, and it's a nice find. The materials here are drawn from four courses at the school: Math 105, Math 106, Math 205, and Math 206. The first couple of resources in each section contain past quizzes and exams from each course, complete with information on each topic. Additionally, each area contains drill problems, tutorials, and a fun "Find the Error!" feature. The topics covered here include linear algebra, quadric surfaces, functions, and abstract vector spaces. Moving on, the site also includes links to external sites from Harvey Mudd College and the University of California-Davis that address advanced math topics. For those persons interested in learning more about the mathematics department at Bates College, there's a link to its official website at the bottom of the page.Mon, 25 Jun 2012 10:53:18 -0500Do the Math
by staff members at the University of Arizona's Center for Recruitment & Retention of Mathematics Teachers (CRR), Do the Math is a weekly cable television show that features mathematics teachers explaining key mathematical concepts. Recently, the folks at CRR decided to create a "best of" playlist that offers segments from this popular program. Here visitors will find 18 segments that last between 26 and 38 minutes. Some of the subjects covered include geometry, advanced algebra, and calculus. Visitors may be interested in the materials on the left-hand side of the page, such as an AP Calculus practice exam, information about the related academic programs offered at the University of Arizona, and more. Also, the site contains a listserv for mathematics teachers and information on upcoming conferences that may be of interest.Thu, 15 Mar 2012 03:00:05 -0500Math Interactives
Interactives is part of a larger site, called LearnAlberta.ca, which is "designed and developed to assist kindergarten to Grade 12 teachers in Alberta locate and utilize digital learning and teaching resources. The design of the site is reflective of how teachers think and work in an online environment." Visitors interested in multiple ways to learn math will love the print and video math activities available on the site. There are four categories from which visitors can choose, on the left hand side of the homepage: Number, Pattern and Relations, Shape and Space and Statistics and Probability. Each section contains a video and an interactive that encourages students to explore the concept in question. For example, students may learn about linear equations through a video about the costs of feeding animals at the Calgary Zoo, and later come up with their own linear equations to predict the costs, accounting for price of food, amount needed, and delivery fees.Wed, 14 Mar 2012 03:00:06 -0500National Council of Teachers of Mathematics: Lessons & Resources
website is an excellent resource for math teachers teaching any age and level of students. The National Council of Teachers of Mathematics (NCTM), "support[s] teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development and research." Visitors interested in making math fun will certainly want to read the lead article on the homepage of the Lessons and Resources section, "Learn Ways to Evaluate Math Games," as it not only helps with game evaluation, but also provides links to good free games. Below the lead article are categories for Elementary, Middle School and High School math. Each category has a publication especially for those grade levels, with current and archived issues available. Also on the homepage of the Lessons and Resources section, visitors will find Family Corner, Teaching Tips, and Lessons and Teaching.Mon, 6 Feb 2012 03:00:03 -0600
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978080922280Real-life examples encourage students to connect math to their own experiences
Full scope of math curriculum in 16 books means students can work on specific areas for review
Every Number Power book targets a particular set of math skills. Students can work on as many or as few concepts as they need. Diagnostic tests and performance-based prescriptions target problem areas. Short, manageable lessons and step-by-step examples assure success. Each book includes quick-reference pages for using calculators, mental math, formulas, measurements, and estimations
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You can type in any algebraic expression, using the operators listed in "Arithmetic". You can use spaces to denote multiplication. Be careful not to forget the space in . If you type in with no space, Mathematica will interpret this as a single symbol, with the name , not as a product of the two symbols and .
Mathematica rearranges and combines terms using the standard rules of algebra.
When you type in more complicated expressions, it is important that you put parentheses in the right places. Thus, for example, you have to give the expression in the form . If you leave out the parentheses, you get instead. It never hurts to put in too many parentheses, but to find out exactly when you need to use parentheses, look at "Operator Input Forms".
When you type in an expression, Mathematica automatically applies its large repertoire of rules for transforming expressions. These rules include the standard rules of algebra, such as , together with much more sophisticated rules involving higher mathematical functions.
The notion of transformation rules is a very general one. In fact, you can think of the whole of Mathematica as simply a system for applying a collection of transformation rules to many different kinds of expressions.
The general principle that Mathematica follows is simple to state. It takes any expression you input, and gets results by applying a succession of transformation rules, stopping when it knows no more transformation rules that can be applied.
• Take any expression, and apply transformation rules until the result no longer changes.
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Modify Your Results
An ideal program for struggling students "Glencoe Algebra: Concepts and Applications" covers all the Algebra 1 concepts. This program is designed for students who are challenged by high school mathematics.
Glencoe Algebra: Concepts and Applications includes lessons that will help students prepare for the Texas Essential Knowledge and Skills assessed on the Texas state test. This textbook contains a special section of practice problems specifically for the Texas state testAlgebra: Concepts & Applications, is a comprehensive Algebra 1 program that is available in full and two-volume editions. Algebra: Concepts & Applicationsuses a clean lesson design with many detailed examples and straightforward narration that make Algebra 1 topics inviting and Algebra 1 content understandable. Volume 1 contains Chapters 1-8 ofAlgebra: Concepts & Applicationsplus an initial section called Chapter A. Chapter A includes a pretest, lessonson prerequisite concepts, and a posttest. Designed for students who are challenged by high school mathematics, the 2007 edition has many new features and support components
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865See below for information about part time tuition fee loans available for study towards a qualification.985There is no real number whose square is –1, but mathematicians long ago invented a system of numbers, called complex numbers, in which the square root of –1 does exist. These complex numbers can be thought of as points in a plane, in which the arithmetic of complex numbers can be pictured. When the ideas of calculus are applied to functions of a complex variable a powerful and elegant theory emerges, known as complex analysis.
The texts have many worked examples, problems and exercises (all with full solutions), and there is a module handbook that includes reference material, the main results and an index. These texts are supported by CDs that teach complex analysis techniques, while another CD presents a discussion of the central role of complex analysis in mathematics. A DVD uses computer graphics to demonstrate many geometric properties of complex functions.
Entry
This is a Level 3 module. Level 3 modules build on study skills and subject knowledge acquired from studies at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with the OU.
There is a diagnostic quiz that will help you to determine whether you are adequately prepared for this module. Your regional centre will also be able to tell you where you can see reference copies of the module units.
Preparatory work
There is no formal preparatory work, but you should revise your algebraic skills, and differential and integral calculus, before the module begins.
Regulations
As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are
available on our Essential documents website.
If you have a disability
The module should present no special difficulties, though it does include a lot of diagrams. The study materials are available on audio in DAISY Digital Talking Book format and there are transcripts of the module audio-visual materialModule books, CDs, DVD.
You will need
CD player and DVD player (or computer able to play DVDs). A scientific calculator would be useful but is not essential.
You require access to the internet at least once a week during the module to download module resources and assignments, and to keep up to date with module news mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module.
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To register a place on this course return to the top of the page and use the Click to register button.
Student Reviews
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"This is an excellent rigorous course. The material is well presented often with a geometric flavour. Basic concepts are presented
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Dickinson, TX Geometry help students develop the ability to see computational problems from a mathematical perspective. Discrete math is normally divided into six areas: sets, functions, and relations; basic logic; proof techniques; counting basics; graphs and trees; and discrete probability. I show students how these topics are interwoven with computer science applications
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Synopses & Reviews
Publisher Comments: performance on standardized tests and state assessment exams. This skill building book addresses all the mathematical concerns 8th graders develop during this critical stage in their academic career. In order to help students achieve success, it provides: A pretest to pinpoint your strengths and weaknesses A posttest to show you the progress you've made 32 short lessons that gradually increase in difficulty Hundreds of practice exercises to help you master essential math skills 8th Grade Math also provides online access to FREE math practice problems, which students can use to: Practice and improve math skills Receive immediate scoring and detailed answer explanations for all questions Benchmark skills and focus study with a customized diagnostic report
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Graham, WA GeGaining facility at understanding how math 'works' is the goal. Once proficient, the concepts of algebra are introduced, with several examples and an effort to understand the vocabulary as well as processes involved. As an instructor, I strive to provide background information and context to my...
...Shopping requires estimating to ensure that you can afford what you are spending and that the bill is correct. Household budgets, selecting a loan, comparing savings and investments require understanding percentages. Plumbers, electricians, and HVAC technicians need to understand volumes, capacities, and flow rates
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Elementary Statistics-Text - 8th edition
Summary: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. In recent editions, Al Bluman has placed more emphasis on conceptual understanding and understanding results, along with increased focus on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing ...show moretechnologies commonly used in such8610393 +$3.99 s/h
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0073386103 Some wear to cover. Text is in great shape. ISBN|0073386103, Elementary Statistics A Step by Step Approach (C.)2012
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On-line lessons and practice for students studying high school level Algebra 2 and Trigonometry for state assessments. Also resources for teachers.
Found several presentations for Answers For Alge2caching Version Of Math Caching Box 7. Preview and Download Answers For Alge2caching Version Of Math Caching Box 7 now.
Browse Ap Biology Campbell Chapter 10 Study Guide Answers websites, images, video and social networks using results from all the most popular search engines on the web ... biology objectives answer.html
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...The topics learned in middle school prealgebra form a foundation of math skills that are used in every math course that comes later in high school and college curricula. This means that if you let your student continue to struggle with fraction arithmetic, arithmetic using negative numbers, fact
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Students calculate the area under a curve. In this calculus lesson plan, students use Riemann sums to find and approximate the area under a curve. They use the derivative and differential equations to solve.
Solve problems using integration and derivatives. By using calculus, learners will analyze graphs to find the extrema and change in behavior. They then will identify the end behavior using the derivatives. Activities and handouts are included.
Young scholars calculate the solution using integrals in this calculus lesson. They use the TI to create a visual of how to compute integrals. Learners also use substitution by integration and all its properties.
Pupils solve problems using integration by parts in this calculus instructional activity. Learners apply the product rule and integration by parts. They graph the equation and use the TI to observe the integration process.
Learners investigate the intervals represented by a function in this calculus lesson. They decide what interval of the function will be positive, negative or zero. They are then given graphs of functions and asked to analyze it.
Students explore scatter functions. In this pre-calculus lesson, students model data, evaluate the function and use their model to interpolate or predict end behavior of the function. The lesson employs the use of a graphing calculator.
Students explore the process of finding the volume of a solid of revolution. In this AP Calculus lesson, students review concepts of finding volumes between two curves before starting this lesson. They observe solids that are formed by the revolutions around a horizontal or vertical axis.
Pupils investigate integrals and their relationship to velocity in this calculus instructional activity. They use the idea of a moving elevator to explore vertical motion and use the TI to help them create a visual.
In this algebra worksheet, students use the summation notation correctly as they solve problems. They define the integral of a function and solve problems involving i. There are 14 questions with an answer key.
In this math worksheet, students complete a series of mathematical investigations to determine readiness for the course of study they are enrolled in. This worksheet is problem set four in a set of worksheets for calculus.
High schoolers engage in daily spelling and vocabulary practice of workplace-related terms with definition matching exercises, word scrambles, dictation, sentence writing. Finally, they compose a short essay in which they use the words correctly.
Students discuss the importance of fundamental theorems in math. In this calculus lesson plan, students define the fundamental theorem of calculus and discuss why it is so important they understand it. They work problems to model how this theorem works.
Humongous Calculus problem of the day - "Two hikers begin ..." Use the properties of indefinite integers and indefinte integrals to solve this problem. Six practice problems are included with space to do the work, as well as a suggestion of ten homework problems from the book.
Students discuss the importance of Integrals as it relates to calculus. In this calculus lesson, students define Integrals and practice taking the integrals of polynomials. They work problems to enhance their understanding of how derivatives is incorporated into calculus.
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The author is an experienced math tutor with a Ph.D. in education and has a California secondary teaching credential in math. She presents questions which a person should use to help select a good math tutor.
An area of vehicle repairs that is something of a mystery to many mechanics, repairing alternators and starter motors is shown in step by step detail in this unique manual. Not only is this ideal for the garage professional,it also offers an opportunity for starting a new and highly profitable business, supplying reconditioned units to the public and garage trade as well as a full repair service.
This book teaches the mechanics and methodology of long division, a procedure for dividing numbers without the need for an electronic calculator. Starting with basic concepts, the book explains the method step by step, and then reinforces these concepts using extensive examples and problems with complete solutions. A Tarrington Math Series Book. Most appropriate for grades 5 to 8.In this article, I suggest how we REALLY learn to spell-and it's not all those boring spelling tests you had in school!
I suggests an approach to children's literacy that is, for a change, productive and makes sense.These six guides are designed for students as introductions to complex philosophical concepts and ideas. Topics covered include deliberative democracy, justice as fairness by Rawls, Aristotles' virtues, Kant's concepts and intuitions, Palto's Symposium, and essay writing strategiesThis guide covers Book 2 of Aristotle's Nicomachean Ethics in which he discusses the virtues. It explains the difference between moral and intellectual virtues, what Aristotle means by 'happiness', what he considers vices, and his 'doctrine of the mean'.
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MathScore®saves you time and is proven to raise math proficiency. MathScore
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Integrated Arithmetic and Basic Algebra - 4th edition
Summary: KEY MESSAGE:Integrated Arithmetic and Basic Algebra, Fourth Edition, integrates arithmetic and algebra to allow students to see the big picture of math. Rather than separating these two subjects, this text helps students recognize algebra as a natural extension of arithmetic. As a result, students see how concepts are interrelated and are better prepared for future courses. KEY TOPICS: Adding and Subtracting Integers and Polynomials; Laws of Exponents, Products and Quotients of ...show moreIntegers and Polynomials; Linear Equations and Inequalities; Graphing Linear Equations and Inequalities; Factors, Divisors, and Factoring; Multiplication and Division of Rational Numbers and Expressions; Addition and Subtraction of Rational Numbers and Expressions; Ratios, Percents, and Applications; Systems of Linear Equations; Roots and Radicals; Solving Quadratic Equations MARKET: For all readers interested in algebra and basic algebra. ...show less
Ships same or next business day with delivery confirmation. Good condition. May or may not contain highlighting. Expedited shipping availableTEXTBOOKFETCHER! Cortland, NY
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Blawenburg CalculusWith the extensive use of analogies and examples, I transform even the most advanced concepts and intricate details into an easily understood and readily recallable framework of knowledge. I emphasize visualization and creating connections between new material and old material to better store in...
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Pre-Calculus for Dummies - 2nd edition
Summary: The. With this guide's help you'll quickly and painlessly get a handle on all of the concepts - not just the number crunching ...show more- and understand how to perform all pre-calc tasks, from graphing to tackling proofs. You'll also get a new appreciation for how these concepts are used in the real world, and find out that getting a decent grade in pre-calc isn't as impossible as you thought.
Updated with fresh example equations and detailed explanations
Tracks to a typical pre-calculus class
Serves as an excellent supplement to classroom learning
If ''the fun and easy way to learn pre-calc'' seems like a contradiction, get ready for a wealth of surprises in Pre-Calculus For Dummies!
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Video Summary: This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations
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Yes, you can...
Master math and science!
Succeed with Math: Every Student's Breaking the Science Barrier: How to
Guide to Conquering Math Anxiety Explore and Understand the Sciences
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The Math Class that's Right for You
Developmental mathematics is a sequence of pre-college level math courses designed to prepare students for college level mathematics.The developmental mathematics program at Weber State University offers Pre-algebra (Math 0950), a First Course in Algebra (Math 0990) and Intermediate Algebra (Math 1010), as well as Pathway to Contemporary Math (Math 810). Many students entering an open enrollment institution like Weber State University need developmental mathematics courses for a variety of reasons.It is the goal of the WSU Developmental Mathematics program to assist students in gaining the math skills they need for success in college level mathematics in as short a time as possible.
Weber State University is a leader in the state when it comes to developmental mathematics reform.Traditional developmental mathematics programs only provide lecture courses, shown by research to be the least effective method of instruction.
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Pathway to Contemporary Math*:
*This course is only for students who plan to take Math 1030, this course will not be a pre-req for any other Math QL course.
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Mathematics - Algebra (529 results)
The purpose of this book, as implied in the introduction, is as follows: to obtain a vital, modern scholarly course in introductory mathematics that may serve to give such careful training in quantitative thinking and expression as well-informed citizens of a democracy should possess. It is, of course, not asserted that this ideal has been attained. Our achievements are not the measure of our desires to improve the situation. There is still a very large "safety factor of dead wood" in this text. The material purposes to present such simple and significant principles of algebra, geometry, trigonometry, practical drawing, and statistics, along with a few elementary notions of other mathematical subjects, the whole involving numerous and rigorous applications of arithmetic, as the average man (more accurately the modal man) is likely to remember and to use. There is here an attempt to teach pupils things worth knowing and to discipline them rigorously in things worth doing.<br><br>The argument for a thorough reorganization need not be stated here in great detail. But it will be helpful to enumerate some of the major errors of secondary-mathematics instruction in current practice and to indicate briefly how this work attempts to improve the situation. The following serve to illustrate its purpose and program:<br><br>1. The conventional first-year algebra course is characterized by excessive formalism; and there is much drill work largely on nonessentials.
Isaac Todhunter's Algebra for Beginners: With Numerous Examples is a mathematics textbook intended for the neophyte, an excellent addition to the library of math instructionals for beginners. Todhunter's textbook has been divided into 44 chapters. Early chapters highlight the most basic principles of mathematics, including sections on the principal signs, brackets, addition, subtraction, multiplication, division, and other topics that form the foundation of algebra. Simple equations make up the large majority of the material covered in this textbook. Later chapters do introduce quadratics, as well as other more advanced subjects such as arithmetical progression and scales of notation. It is important to note that Todhunter sticks very much to the basics of algebra. The content of this book lives up to its title, as this is very much mathematics for beginners. The content is provided in an easy to follow manner. This book could thus be used for independent learning as well as by a teacher. A great deal of focus has clearly been given to providing examples. Each concept is accompanied by numerous sample questions, with answers provided in the final chapter of the book. The example questions are every bit as important as the explanations, as one cannot begin to grasp mathematical concepts without having the opportunity to put them into practice. The basics of algebra are explained in an easy to follow manner, and the examples provided are clear and help to expand the knowledge of the learner. If given a chance, Isaac Todhunter's Algebra for Beginners: With Numerous Examples can be a valuable addition to your library of mathematics textbooks.
Bertrand Russell was a British logician, nobleman, historian, social critic, philosopher, and mathematician. Known as one of the founders of analytic philosophy, Russell was considered the premier logician of the 20th century and widely admired and respected for his academic work. In his lifetime, Russell published dozens of books in wildly varying fields: philosophy, politics, logic, science, religion, and psychology, among which The Principles of Mathematics was one of the first published and remains one of the more widely known. Although remembered most prominently as a philosopher, he identified as a mathematician and a logician at heart, admitting in his own biography that his love of mathematics as a child kept him going through some of his darkest moments and gave him the will to live. With his book The Principles of Mathematics, Russell aims to instill the same deep seated passion for mathematics and logic that he has carefully cultivated in the reader. He adeptly explores mathematical problems in a logical context, and attempts to prove that the study of mathematics holds critical importance to philosophy and philosophers. Russell utilizes the text to explore the some of the most fundamental concepts of mathematics, and expounds on how these building blocks can easily be applied to philosophy. In the second part of the book, Bertrand addresses mathematicians directly, discussing arithmetic and geometry principles through the lens of logic, offering yet another unique and groundbreaking interpretation of a field long before considered static. This book affords new insight and application for many basic mathematical concepts, both in roots of and application to other fields of scholarly pursuit. Russell uses his book to establish a baseline of mathematical understanding and then expands upon that baseline to establish larger and more complex ideas about the world of mathematics and its connections to other fields of personal interest. The Principles of Mathematics is a very captivating glimpse into the logic and rational of one of history's greatest thinkers. Whether you're a mathematician at heart, a logician, or someone interested in the life and thoughts of Bertrand Russell, this book is for you. With an incredible amount of information on mathematics, philosophy, and logic, this text inspires the reader to learn more and discover the ways in which these very disparate fields can interconnect and create new possibilities at their intersections.
Florian Cajori's A History of Mathematics is a seminal work in American mathematics. The book is a summary of the study of mathematics from antiquity through World War I, exploring the evolution of advanced mathematics. As the first history of mathematics published in the United States, it has an important place in the libraries of scholars and universities. A History of Mathematics is a history of mathematics, mathematicians, equations and theories; it is not a textbook, and the early chapters do not demand a thorough understanding of mathematical concepts. The book starts with the use of mathematics in antiquity, including contributions by the Babylonians, Egyptians, Greeks and Romans. The sections on the Greek schools of thought are very readable for anyone who wants to know more about Greek arithmetic and geometry. Cajori explains the advances by Indians and Arabs during the Middle Ages, explaining how those regions were the custodians of mathematics while Europe was in the intellectual dark ages. Many interesting mathematicians and their discoveries and theories are discussed, with the text becoming more technical as it moves through Modern Europe, which encompasses discussion of the Renaissance, Descartes, Newton, Euler, LaGrange and Laplace. The final section of the book covers developments in the late 19th and early 20th Centuries. Cajori describes the state of synthetic geometry, analytic geometry, algebra, analytics and applied mathematics. Readers who are not mathematicians can learn much from this book, but the advanced chapters may be easier to understand if one has background in the subject matter. Readers will want to have A History of Mathematics on their bookshelves.
Bringing to life the joys and difficulties of mathematics this book is a must read for anyone with a love of puzzles, a head for figures or who is considering further study of mathematics. On the Study and Difficulties of Mathematics is a book written by accomplished mathematician Augustus De Morgan. Now republished by Forgotten Books, De Morgan discusses many different branches of the subject in some detail. He doesn't shy away from complexity but is always entertaining. One purpose of De Morgan's book is to serve as a guide for students of mathematics in selecting the most appropriate course of study as well as to identify the most challenging mental concepts a devoted learner will face. "No person commences the study of mathematics without soon discovering that it is of a very different nature from those to which he has been accustomed," states De Morgan in his introduction. The book is divided into chapters, each of which is devoted to a different mathematical concept. From the elementary rules of arithmetic, to the study of algebra, to geometrical reasoning, De Morgan touches on all of the concepts a math learner must master in order to find success in the field. While a brilliant mathematician in his own right, De Morgan's greatest skill may have been as a teacher. On the Study and Difficulties of Mathematics is a well written treatise that is concise in its explanations but broad in its scope while remaining interesting even for the layman. On the Study and Difficulties of Mathematics is an exceptional book. Serious students of mathematics would be wise to read De Morgan's work and will certainly be better mathematicians for it.
The present work is intended as a sequel to our Elementary Algebra for Schools. The first few chapters are devoted to a fuller discussion of Ratio, Proportion, Variation, and the Progressions, which in the former work were treated in an elementary manner; and we have here introduced theorems and examples which are unsuitable for a first course of reading.<br><br>From this point the work covers ground for the most part new to the student, and enters upon subjects of special importance: these we have endeavoured to treat minutely and thoroughly, discussing both bookwork and examples with that fulness which we have always found necessary in our experience as teachers.<br><br>It has been our aim to discuss all the essential parts as completely as possible within the limits of a single volume, but in a few of the later chapters it has been impossible to find room for more than an introductory sketch; in all such cases our object has been to map out a suitable first course of reading, referring the student to special treatises for fuller information.<br><br>In the chapter on Permutations and Combinations we are much indebted to the Rev. W. A. Whitworth for permission to make use of some of the proofs given in his Choice and Chance.
There are many men and women who, from lack of opportunity or some other reason, have grown up in ignorance of the elementary laws of science. They feel themselves continually handicapped by this ignorance. Their critical faculty is eager to submit, alike old established beliefs and revolutionary doctrines, to the test of science. But they lack the necessary knowledge.<br><br>Equally serious is the fact that another generation is at this moment growing up to a similar ignorance. The child, between the ages of six and twelve, lives in a wonderland of discovery; he is for ever asking questions, seeking explanations of natural phenomena. It is because many parents have resorted to sentimental evasion in their replies to these questionings, and because children are often allowed either to blunder on natural truths for themselves or to remain unenlightened, that there exists the body of men and women already described. On all sides intelligent people are demanding something more concrete than theory; on all sides they are turning to science for proof and guidance.<br><br>To meet this double need - the need of the man who would teach himself the elements of science, and the need of the child who shows himself every day eager to have them taught him - is the aim of the "Thresholds of Science" series.<br><br>This series consists of short, simply written monographs by competent authorities, dealing with every branch of science - mathematics, zoology, chemistry and the like. They are well illustrated, and issued at the cheapest possible price.
This text is prepared to meet the needs of the student who will continue his mathematics as far as the calculus, and is written in the spirit of applied mathematics. This does not imply that algebra for the engineer is a different subject from algebra for the college man or for the secondary student who is prepared to take such a course. In fact, the topics Avhich the engineer must emphasize, such as numerical com)utations, checks, graphical methods, use of tables, and the solution of specific problems, are among the most vital features of the subject for any student. But important as these topics are, they do not comprise the substance of algebra, which enables it to serve as part of the foundation for future work. Rather they furnish an atmosphere in which that foundation may be well and intelligently laid. The concise review contained in the first chapter covers the topics which have direct bearing on the work which follows. No attempt is made to repeat all of the definitions of elementary algebra. It is assumed that the student retains a certain residue from his earlier study of the subject. The quadratic equation is treated with unusual care and thoroughness. This is done not only for the purpose of review, but because a mastery of the theory of this equation is absolutely necessary for effective work in analytical geometry and calculus. Furthermore, a student who is well grounded in this particular is in a position to appreciate the methods and results of the theory of the general equation with a minimum of eii ort. The theory of equations forms the keystone of most courses in higher algebra. The chapter on this subject is developed gradually, and yet with pointed directness, in the hope that the processes which students often perform in a perfunctory manner will take on additional life and interest.
The present work contains a full and complete treatment of the topics usually included in an Elementary Algebra. The author has endeavored to prepare a course sufficiently advanced for the best High Schools and Academies, and at the same time adapted to the requirements of those who are preparing for admission to college.<br><br>Particular attention has been given to the selection of examples and problems, a sufficient number of which have been given to afford ample practice in the ordinary processes of Algebra, especially in such as are most likely to be met with in the higher branches of mathematics. Problems of a character too difficult for the average student have been purposely excluded, and great care has been taken to obtain accuracy in the answers.<br><br>The author acknowledges his obligations to the elementary text-books of Todhunter and Hamblin Smith, from which much material and many of the examples and problems have been derived. He also desires to express his thanks for the assistance which he has received from experienced teachers, in the way of suggestions of practical value.
The Directly Useful Technical Series requires a few words by way of introduction. Technical books of the past have arranged themselves largely under two sections: the theoretical and the practical. Theoretical books have been written more for the training of college students than for the supply of information to men in practice, and have been greatly filled with problems of an academic character. Practical books have often sought the other extreme, omitting the scientific basis upon which all good practice is built, whether discernible or not. The present series is intended to occupy a midway position. The information, the problems, and the exercises are to be of a directly useful character, but must at the same time be wedded to that proper amount of scientific explanation which alone will satisfy the inquiring mind. We shall thus appeal to all technical people throughout the land, either students or those in actual practice.
Already Published Anthropology By R.R. Makett An Introduction To Science By J.Arthuk Tnousos Evolution By J, Abthuk Thohson am The Animal World By F.W. Gakele Introduction To Mathe-Matics By A.N. Whitehead Astronomy By A. R.Hinks Psychical Research. By W.F. Eabbett The Evolution Of Plants By D.H. Scott Crime And Insanity. By C.A. Mebcieb Matter And Energy. By F.Sodd Psychology By W.McDouoau. Principles Of Physiology By J.G. McKendrick The Making Of The Earth By J.W. Gregoev Electricity By Gisbest Kapp The Human Body By A.Kiitb Future Issues Chemistry By R.Meldola The Mineral World. By SiT.
The work on Algebra of which this volume forms the first part, is so far elementary that it begins at the beginning of the subject. It is not, however, intended for the use of absolute beginners. The teaching of Algebra in the earlier stages ought to consist in a gradual generalisation of Arithmetic; in other words, Algebra ought, in the first instance, to be taught as Arithmetica Universalis in the strictest sense. I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with the construction of literal formulae, such, for example, as that for the amount of a sum of money during a given term at simple interest Then it becomes necessary, if Algebra is to be anything more than a mere bundle of unconnected rules, to lay down generally the three fundamental laws of the subject, and to proceed deductively in short, to introduce the idea of Algebraic Form, which is the foundation of all the modern developments of Algebra and the secret of analytical geometry, the most beautiful of all its applications.
This tract is intended to give an account of the theory of equations according to the ideas of Galois. The conspicuous merit of this method is that it analyses, so far as exact algebraical processes permit, the set of roots possessed by any given numerical equation. To appreciate it properly it is necessary to bear constantly in mind the difference between equalities in value and identities or equivalences in form; I hope that this has been made sufficiently clear in the text. The method of Abel has not been discussed, because it is neither so clear nor so precise as that of Galois, and the space thus gained has been filled up with examples and illustrations.<br><br>More than to any other treatise, I feel indebted to Professor H. Weber's invaluable Algebra, where students who are interested in the arithmetical branch of the subject will find a discussion of various types of equations, which, for lack of space, I have been compelled to omit.<br><br>I am obliged to Mr Morris Owen, a student of the University College of North Wales, for helping me by verifying some long calculations which had to be made in connexion with Art. 52.
The subject-matter of this book is a historical summary of the development of mathematics, illustrated by the lives and discoveries of those to whom the progress of the science is mainly due. It may serve as an introduction to more elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know.<br><br>The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics. In the second edition, issued in 1893, I rearranged parts of it, and introduced a good deal of additional matter. The third edition, issued in 1901, was revised, but not materially altered; and the present edition is practically a reprint of this, save for a few small corrections and additions.
This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission:<br><br>A. Makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately owned rights; or<br><br>B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report.<br><br>As used in the above, "person acting on behalf of the Commission" includes any employee or contractor of the Commission to the extent that such employee or contractor prepares, handles or distributes, or provides access to, any information pursuant to his employment or contract with the Commission.
Edward Everett Whitford was bom in Brookfield, N.Y., January 31, 1865;graduated from Brookfield Academy in 1881; received the degree of A.B. from Colgate University in 1886 and of A.M. in 1890. He taught in Colbj Academy, New London, N.H., Keystone Academy, Factorjrville, Pa., Shamokin (Pa.) High School, Commercial High School and Pratt Institute, Brooklyn, N.Y. He was principal of Brookfield High School, 1900-1. He is now instructor in mathematics in the College of the City of New York with which institution he has beei connected since 1905. He has been a graduate student in Coliunbia University since February, 1904, and is a member of the American Mathematical Society. The writer takes this opportunity of expressing hithanks to Professor David Eugene Smith for fruitfnsuggestions and able and helpful criticism.523 West ISIbt Str., New Yobk, December, 1911.
The Principles of Mathematics: Vol. 1 is a terrific introduction to the fundamental concepts of mathematics. Although the book's title involves mathematics, it is not a textbook packed with equations and theorems. Instead philosopher Bertrand Russell uses mathematics to explore the structure of logic. Russell's ultimate point is that mathematics is logic and logic itself is truth. The book is substantial and covers all subjects of mathematics. It is divided into seven sections: indefinables in mathematics, number, quantity, order, infinity and continuity, space, matter and motion. Russell covers all the major developments of mathematics and the contributions of important figures to the field. His sharp mind is evident throughout The Principles of Mathematics, as he challenges established rules and teachers readers how to think through difficult problems using logic. Russell was one of the great minds of the 20th Century. In this book he discusses how his ideas were influenced by the logician Peano. He also debates other philosophers and mathematicians, and even anticipates the Theory of Relativity, which had not yet been published by Einstein. One does not need to love mathematics to gain insights from The Principles of Mathematics: Vol. 1. Those who are interested in logic, intellectualism, philosophy or history will find significant insights into logical principles. Readers who desire an intellectual challenge will truly enjoy The Principles of Mathematics: Vol. 1.
Counting a series of things and keeping tally of the tens on the fingers were processes used by primitive peoples. From the ten fingers arose ultimately the decimal system of numeration. Recording the results of counting was done by the Egyptians and other ancient nations by means of strokes and hooks; for one thing a single stroke | was made, for two things two strokes || were used, and so on up to ten which was represented by Π. Then eleven was written |Π, twelve ||Π, and so on up to twenty, or two tens, which was represented by ΠΠ. In this way the numeration proceeded up to a hundred, for which another symbol was employed.<br><br>Names for ||, |||, ||||, ΠΠ, etc., appear in the Egyptian hieroglyphics, but a special symbol for each name is not used. Probably the Hindoos first invented such symbols, and passed them on to the Arabs, through whom they were introduced into Europe.<br><br>2<br><br>Greek Notation<br><br>The Greeks used an awkward notation for recording the results of counting.
In preparing the present work, the author has endeavored to meet the needs of colleges and scientific schools of the highest rank.<br><br>The development of the subject follows in the main the author's College Algebra; but numerous improvements have been introduced.<br><br>Attention is especially invited to the following:<br><br>1. The development of the fundamental laws of Algebra for the positive and negative integer, the positive and negative fraction, and zero, in Chaps. I and II.<br><br>In the above treatment, the author has followed to a certain extent The Number System of Algebra, by Professor H. B. Fine; who has very courteously permitted this use of his treatise.<br><br>2. The development of the principles of equivalence of equations, and systems of equations, both linear and of higher degrees; see 116-123, 182, 233-6, 396, 442, 470, 477, and 478.<br><br>3. The prominence given to graphical representation.<br><br>In Chap. XIV, the student learns how to obtain the graphs of linear equations with two unknown numbers, and of linear expressions with one unknown number. He also learns how to represent graphically the solution of a system of two linear equations, involving two unknown numbers, and sees how indeterminate and inconsistent systems are represented graphically.<br><br>The graphical representation of quadratic expressions, with one unknown number, is taken up in 465; and, in 467, the graphical representation of equal and imaginary roots.
This text differs widely from that marked out by custom and tradition. It treats the various branches of mathematics more with reference to their unities and less as isolated entities (sciences). It seeks to give pupils usable knowledge of the principles underlying mathematics and ready control of them. These texts are not an experiment; they were thoroughly tried out in mimeograph form on hundreds of high school pupils before being put into book form. The scope of Books I and II does not vary greatly from that covered in algebras and geometries of the usual type. However, Book I is different in that arithmetic, algebra, and geometry are treated side by side. The effect of this arrangement is increased interest and power of analysis on the part of the learner, and greater accuracy in results. Some pupils like arithmetic, others like algebra, still others like geometry; the change is helpful in keeping up interest. The study of geometry forces analysis at every step and stage; consequently written problems and problems to be stated have no terrors for those who are taught in this way. For several years mathematical associations have urged that all work should be based upon the equation. In accordance with this view we have made the demonstrations in this book largely algebraic, thus making the demonstration essentially a study in simultaneous equations. In this method of treatment, we have found it advantageous not to hurry the work. Pupils gain power, not in solving many problems, but in analyzing and tio?oxt 3 xaAwafcaxs.- ing the principles of a few.
The object of what are here called mathematical essays is to co-ordinate a pupils knowledge on certain subjects which are not specially dealt with in text-books. The essays, of which outlines are given in the first part of this book, are of the following types: (i) A group of theorems on one subject, the theorems in ordinary text-books being often scattered in one or several volumes (e.g., Essays 19, 26, 70, 91);(ii) A series of questions leading up to the solution of an important problem (e.g., Essays 79, 87, 90, 93);(iii) A collection of different methods of proving the same theorem (e.g., Essays 28, 61, 75, 78);(iv) A series of applications- of the same theorem (e.g., Essays 37, 48, 75); (v)A classification of tests of the same geometrical condition (e.g. Essays 3, 4, 40, 41).The subjects given in the first part (Essays1-100) are, as a rule, of an elementary character. In several of these, a question, which throws light on different subjects, is repeated, Those given in the second part (Essays101-200) are taken from papers set for entrance scholarships in the Trinity and Pembroke groups of Cambridge colleges from 1905 and 1907 respectively. I should be grateful for notices of any errors that may be found in the text or answers, and for any suggestions from teachers for the improvement of the book. Charles Davison. Birmingham, November, 1914.
Teacher's Manual for First-Year Mathematics is a book written by George William Myers, a Professor of the Teaching of Mathematics and Astronomy at the University of Chicago. The book is intended as a teaching manual for teachers instructing their students using a textbook called First Year Mathematics. Myers' book is intended as a companion piece to the textbook First Year Mathematics, released by the same publishing company, The University of Chicago Press. The book makes effort to assist the teacher by providing them with a detailed how-to regarding teaching the specific problems presented in the textbook. Teacher's Manual is presented in chapters, each corresponding to a chapter in First Year Mathematics. Specific references are made to page numbers and problems presented in the textbook. In total, the book contains fourteen different chapters. Teacher's Manual for First-Tear Mathematics can only be used in conjunction with the appropriate textbook. Without access to First Year Mathematics, the book is of no use. It is however an excellent companion piece to the textbook, and those able to access the original textbook will surely find this text to be highly beneficial. While a well-written teacher's manual, George William Myers' book assumes the reader has access to the original textbook. If you are interested in making use of this manual, do ensure that you are also able to access First Year Mathematics.
It is the purpose of this work to present a through investigation of the various systems of Symbolic Reasoning allied to ordinary Algebra. The chief examples of such systems are Hamilton's Quaternions, Grassmann's Calculus of Extension and Boole's Symbolic Logic. Such algebras have an intrinsic value for separate detailed study; also they are worthy of a comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular.<br><br>The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge. Accordingly after the general principles of the whole subject have been discussed in Book I. of this volume, the remaining books of the volume are devoted to the separate study of the Algebra of Symbolic Logic, and of Grassmann's Calculus of Extension, and of the ideas involved in them. The idea of a generalized conception of space has been made prominent, in the belief that the properties and operations involved in it can be made to form a uniform method of interpretation of the various algebras.<br><br>Thus it is hoped in this work to exhibit the algebras both as systems of symbolism, and also as engines for the investigation of the possibilities of thought and reasoning connected with the abstract general idea of space. A natural mode of comparison.between the algebras is thus at once provided by the unity of the subject-matters of their interpretation. The detailed comparison of their symbolic structures has been adjourned to the second volume, in which it is intended to deal with Quaternions, Matrices, and the general theory of Linear Algebras. This comparative anatomy of the subject was originated by B. Peirce's paper on Linear Associative Algebra, and has been carried forward by more recent investigations in Germany.
This text presents a course in elementary mathematics adapted to the needs of students in the freshman year of an ordinary college or technical school course, and of students in the first year of a junior college. The material of the text includes the essential and vital features of the work commonly covered in the past in separate courses in college algebra, trigonometry, and analytical geometry.<br><br>The fundamental idea of the development is to emphasize the fact that mathematics cannot be artificially divided into compartments with separate labels, as we have been in the habit of doing, and to show the essential unity and harmony and interplay between the two great fields into which mathematics may properly be divided; viz., analysis and geometry.<br><br>A further fundamental feature of this work is the insistence upon illustrations drawn from fields with which the ordinary student has real experience. The authors believe that an illustration taken from life adds to the cultural value of the course in mathematics in which this illustration is discussed. Mathematics is essentially a mental discipline, but it is also a powerful tool of science, playing a wonderful part in the development of civilization. Both of these facts are continually emphasized in this text and from different points of approach.<br><br>The student who has in any sense mastered the material which is presented will at the same time, and without great effort, have acquired a real appreciation of the mathematical problems of physics, of engineering, of the science of statistics, and of science in general.<br><br>A distinctly new feature of the work is the introduction of series of "timing exercises" in types of problems in which the student may be expected to develop an almost mechanical ability. The time which is given in the problems is wholly tentative; it is hoped, in the interest of definite and scientific knowledge concerning what may be expected of a freshman, that institutions using this text will keep a somewhat detailed record of the time actually made by groups of their students.
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Advanced Algebra II: Conceptual Explanations
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534469 / ISBN-13: 9780198534464
Visual Complex Analysis
This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the ...Show synopsisThis radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.Hide synopsis
Visual Complex Analysis
This book is an exceptional example of the way a fully realized alternate approach to a body of knowledge can reveal new truths about that knowledge and, even more important, a deeper understanding of new ways of approaching knowledge. One of the best books on mathematics I know, and that's not just ...
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The current crop of mathematicians are growing up in a more "visual" world and while the different texts of Brown/Churchill and Bressoud are very good, Needham does an excellent job with visualization of the principles and showing how the pieces fit together from a time-line perspective
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MATHEMATICS 091
Intermediate Algebra, Individualized - Module 1 (1)
On-campus computer-based instruction. Simplifying and evaluating rational expressions, solving equations involving rational expressions, applications. This is the 1st92
Intermediate Algebra, Individualized - Module 2 (1)
On-campus computer-based instruction. Functions, linear models and graphs, solving linear inequalities, solving absolute value equations and inequalities. The 291 is required. Mandatory P/NC grading.
MATHEMATICS 093
Intermediate Algebra, Individualized - Module 3 (1)
On-campus computer-based instruction. Using rational exponents, simplifying radical expressions, and solving radical equations. This is the 3or higher on the Algebra COMPASS test, or a score of 28 or higher on ther College Algebra COMPASS test. Previous credit for or concurrent enrollment in MATH 092 is required. Mandatory P/NC grading.
MATHEMATICS 094
Intermediate Algebra, Individualized - Module 4 (1)
On-campus computer-based instruction. Solving quadratic equations by completing the square & the quadratic formula, graphing quadratic functions, applications. The 4th btter), or a score of 51 or higher on the Algebra COMPASS test, or a score of 28 or higher on the College Algebra COMPASS test. Previous credit for or concurrent enrollment in MATH 093 is required. Mandatory P/NC grading.
MATHEMATICS 095
Intermediate Algebra, Individualized - Module 5 (1)
On-campus computer-based instruction. Evaluating exponential and logarithmic expressions, solving these types of equations. This is the 5th in a sequence of 5 self-paced courses, MATH 091-095. The 5-course sequence is equivalent to MATH 099. Students must complete the entire MA94 is required. Mandatory P/NC grading.
MATHEMATICS 096
Intermediate Algebra, Individualized - Modules (5)
On-campus computer-based sequence of 1-credit self-paced modules. MATH 091-095. Rational, radical, and quadratic expressions and equations. Introduction to functions: linear, quadratic, exponential, and logarithmic. MATH 081-085 and MATH 091-095 together are equivalent to MATH 080 and MATH 099. Must be completed within one year. Prerequisite: MATH 080 (2.0 or better),99
Intermediate Algebra II (0)
Simplifying and evaluating linear, quadratic, polynomial, radical, rational, exponential, and logarithmic expressions. Solving these same types of equations and inequalities with graphs and applications to real world modeling and investigating functions. Prerequisite: MATH 098 (2.0 or better), or a score of 51 or higher on the Algebra Compass test, or a score of 28 or higher on the College Algebra COMPASS test. Student Option Grading.
MATHEMATICS & 107
Math in Society (5)
Practical applications of mathematics as they arise in everyday life.Includes finance math, probability & statistics, and a selection of other topics. Designed for students who are not preparing for calculus.Prerequisite: MATH 098 (2.0 or better), MATH 099 (2.0 or better), MATH 095 (2.0 or better), or a score of 69 or higher on the Algebra COMPASS test or a score of 35 or higher on the College Algebra COMPASS test. Student Option Grading.
MATHEMATICS 111
Elements of Pre-Calculus (5)
Algebra topics including mathematical modeling, graphing & problem solving w/ polynomial, rational, exponential & logarithmic functions. Applications. Topics from mathematics of finance. Intended for students in business, social sciences & some biological sciences 141
Precalculus I (5)
The elementary functions and their graphs, with applications to mathematical modeling. Examples include linear, quadratic, polynomial, rational, exponential and logarithmic functions, composite functions, inverse functions and transformation of graphs 146
Introduction to Stats (5)
Analysis of data through graphical and numerical methods, linear regression, the Normal distribution, data collection, elementary probability, confidence intervals and hypothesis testing. Emphasis on applications. Prerequisite: MATH 098 (2.0 or better), MATH 099 (2.0 or better),MATH 095 (2.0 or better), or a score of 69 or higher on the Algebra COMPASS test or a score of 35 or higher on the College Algebra COMPASS test, AND placement in ENGL 100 or ESL 100. Student Option Grading.
MATHEMATICS & 148
Business Calculus (5)
Differential and Integral Calculus of elementary functions with an emphasis on business and social science applications. Designed for students who want a brief course in Calculus. (No credit given to those who have completed MATH& 151.) Prerequisite: MATH 111 preferred (2.0 orbetter) or MATH& 141 (2.0 or better), or a score of 70 or higher on the College Algebra COMPASS test. Student option grading.
MATHEMATICS & 151
Calculus I (5)
Definition, interpretation and applications of the derivative. Derivatives of algebraic and transcendental functions. Prerequisite: MATH& 142 (2.0 or better), or a score of 70 or higher on both the College Algebra and Trigonometry COMPASS Tests. Student option grading.
MATHEMATICS & 171
Math for Elem Ed I (5)
Fundamental concepts of numbers and operations related to topics taught at the K-8 level. Topics include problem solving, algebraic thinking, numberation, and arithmetic with rational numbers. Recommended for future elementary teachers. Prerequisite: MATH 098 (2.0 or better), MATH 099 (2.0 or better), or MATH 095 (2.0 or better), or a score of 69 or higher on the Algebra COMPASS test or a score of 35 or higher on the College Algebra COMPASS test, AND placement into ENGL 100 or ESL 100. Student Option Grading.
MATHEMATICS 297
Individual Project in Mathematics (1)
Individual project in a specific area of mathematics. By arrangement with instructor. Prerequisite: Instructor permission, based on evaluation of student's educational and work experience. Student option grading.
MATHEMATICS 298
Individual Project in Mathematics (2)
Individual project in a specific area of mathematics. By arrangement with instructor. Prerequisite: Instructor permission, based on evaluation of student's educational and work experience. Student option grading.
MATHEMATICS 299
Individual Project in Mathematics (3)
Individual project in a specific area of mathematics, by arrangement with instructor. Prerequisite: Instructor permission based on evaluation of student's educational and work experience. Student option grading.
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Now available! The Information Literacy User's Guide
Good researchers have a host of tools at their disposal that make navigating today's complex information ecosystem much more manageable. Gaining the knowledge, abilities, and self-reflection necessary to be a good researcher helps not only in academic settings, but is invaluable in any career, and throughout one's life. The Information Literacy User's Guide will start you on this route to success.
The Information Literacy User's Guide is based on two current models in information literacy: The 2011 version of The Seven Pillars Model, developed by the Society of College, National and University Libraries in the United Kingdom and the conception of information literacy as a metaliteracy, a model developed by one of this book's authors in conjunction with Thomas Mackey, Dean of the Center for Distance Learning at SUNY Empire State College. These core foundations ensure that the material will be relevant to today's students.
The Information Literacy User's Guide introduces students to critical concepts of information literacy as defined for the information-infused and technology-rich environment in which they find themselves. This book helps students examine their roles as information creators and sharers and enables them to more effectively deploy related skills. This textbook includes relatable case studies and scenarios, many hands-on exercises, and interactive quizzes.
Real Analysis Textbook
How We got from There to Here: A Story of Real Analysis
Robert Rogers, Eugene Boman
The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis. This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context. However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.
Dr. Diane Kiernan, SUNY ESF
Natural Resources Biometrics
Diane Kiernan
Natural Resources Biometrics begins with a review of descriptive statistics, estimation, and hypothesis testing. The following chapters cover one- and two-way analysis of variance (ANOVA), including multiple comparison methods and interaction assessment, with a strong emphasis on application and interpretation. Simple and multiple linear regressions in a natural resource setting are covered in the next chapters, focusing on correlation, model fitting, residual analysis, and confidence and prediction intervals. The final chapters cover growth and yield models, volume and biomass equations, site index curves, competition indices, importance values, and measures of species diversity, association, and community similarity.
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A Closer Look at Tests of Significance -- Which Test?, A Closer Look at Tests of Significance Algebra I Second Edition is a clear presentation of algebra for the high school student. Volume 1 includes the first 6 chapters and covers the following topics: Equations and Functions, Real Numbers, Equations of Lines, Graphs of Equations and Functions, Writing Linear Equations, and Linear Inequalities 1 includes the first Algebra FlexBook is an introduction to algebraic concepts for the high school student. Topics include: Equations & Functions, Real Numbers, Equations of Lines, Solving Systems of Equations & Quadratic Equ Probability and Statistics – A Short Course is an introduction to theoretical probability and data organization. Students learn about events, conditions, random variables, and graphs and tables that allow them to manage data Geometry FlexBook is a clear presentation of the essentials of geometry for the high school student. Topics include: Proof, Congruent Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transform 2 includes the last Advanced Probability and Statistics-Second Edition is a clear presentation of the basic topics in statistics and probability, but finishes with the rigorous topics an advanced placement course requires. Volume 1 includes the first 6 chapters and covers the following topics: Analyzing Statistical Data, Visualizations of Data, Discrete Probability Distribution, Normal Distribution, and Experimentation Algebra, Volume 2 Of 2 FlexBook covers the following six chapters:Systems of Equations and Inequalities; Counting Methods - introduces students to linear systems of equations and inequalities as well as probability and combinations. Operations on linear systems are covered, including addition, subtraction, multiplication, and division. Exponents and Exponential Functions - covers more complex properties of exponents when used in functions. Exponential decay and growth are considered, as are geometric sequences and scientific notation. Polynomials and Factoring; More on Probability - introduces students to polynomials and their basic operations as well as the process of factoring polynomials, quadratic expressions, and special products. Also considered is probability through compound events. Quadratic Equations and Functions - introduces students to quadratic equations and various methods of solving them. Also considered is the discriminant and linear, exponential, and quadratic models. Radicals and Geometry Connections; Data Analysis - covers the concept of the radical and its uses in geometry, including the Distance and Midpoint Formula and Pythagorean's Theorem. Also considered are methods of analyzing data with charts and graphs. Rational Equations and Functions; Statistics - covers rational functions and the operations of rational expressions. Students learn to graph rational functions, divide polynomials, and analyze surveys and samples.'
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reflects more than 25 years of author involvement with business math education and the business community. The focus of this edition is on linking mathematics with real business practices in real businesses--giving readers a better appreciation for and understanding of the concepts that are vital in the business world.The book is filled with
Table of Contents
1. Whole Numbers. Place Value and Our Number System. Read whole numbers. Write whole numbers. Round whole numbers. Operations with Whole Numbers-Five-Step Problem-Solving Strategy introduced with addition and continued throughout text. Add whole numbers. Subtract whole numbers. Multiply whole numbers. Divide whole numbers. 2. Fractions. Fractions. Identify types of fractions. Convert an improper fraction to a whole or mixed number. Convert a whole or mixed number to an improper fraction. Reduce a fraction to lowest terms. Raise a fraction to higher terms. Adding and Subtracting Fractions. Add fractions with like (common) denominators. Find the least common denominator for two or more fractions. Add fractions and mixed numbers. Subtract fractions and mixed numbers. Multiplying and Dividing Fractions. Multiply fractions and mixed numbers. Divide fractions and mixed numbers. 3. Decimals. Decimals and the Place-Value System. Read and write decimals. Round decimals. Operations with Decimals. Add and subtract decimals. Multiply decimals. Divide decimals. Decimal and Fraction Conversions. Convert a decimal to a fraction. Convert a fraction to a decimal. 4. Banking. Checking Account Forms. Make account transactions. Record account transactions. Bank Statements. Reconcile a bank statement with an account register. 5. Equations. Equations. Solve equations using multiplication or division. Solve equations using addition or subtraction. Solve equations using more than one operation. Solve equations containing multiple unknown terms. Solve equations containing parentheses. Solve equations that are proportions. Using Equations to Solve Problems. Use the problem-solving approach to analyze and solve word problems. 6. Percents. Percent Equivalents. Write a whole number, fraction, or decimal as a percent. Write a percent as a whole number, fraction, or decimal. Solving Percentage Problems. Identify the rate, base, and portion in percent problems. Use the percentage formula to find the unknown value when two values are known. Increases and Decreases. Find the amount of increase or decrease in percent problems. Find the new amount directly in percent problems. Find the rate or the base in increase or decrease problems. 7. Business Statistics. Measures of Central Tendency. <
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Marina Del Rey Physics Chicago, as part of my coursework in graduate school for my Master's degree, we had to use linear algebra and have an advanced knowledge of it to understand astronomy, cosmology, and astrophysics. Star behavior and other astronomical calculations are dependent on the use of
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More About
This Textbook
Overview
Mathematical reform is the driving force behind the organization and development of this new precalculus text. The use of technology,primarily graphing utilities,is assumed throughout the text. The development of each topic proceeds from the concrete to the abstract and takes full advantage of technology,wherever appropriate. The first major objective of this book is to encourage students to investigate mathematical ideas and processes graphically and numerically,as well as algebraically. Proceeding in this way,students gain a broader,deeper,and more useful understanding of a concept or process. Even though concept development and technology are emphasized,manipulative skills are not ignored,and plenty of opportunities to practice basic skills are present. A brief look at the table of contents will reveal the importance of the function concept as a unifying theme.
The second major objective of this book is the development of a library of elementary functions,including their important properties and uses. Having this library of elementary functions as a basic working tool in their mathematical tool boxes,students will be able to move into calculus with greater confidence and understanding. In addition,a concise review of basic algebraic concepts is included in Appendix A for easy reference,or systematic review.
The third major objective of this book is to give the student substantialexperience in solving and modeling real world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful. Most of the applications are simplified versions of actual real-world problems taken from professional journalsandprofessional books. No specialized experience is required to solve any of the
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Basic Algebra
Basic Algebra topics are the very beginning of learning Algebra. When you start on these topics, you're expected to have little more than Arithmetic knowledge (you know, those problems to do with "+", "-", "÷", and "×"). You'll also need to be familiar with Pre Algebra topics, but you don't need to worry about that now. You can easily refer back to the Pre Algebra topics later if you need to.
This section is designed specifically for those who are beginning Algebra, but will be useful for reference if you're learning Intermediate or Advanced algebra topics. You'll find a list of topics followed by a brief description.
How do you use these pages? If you're starting out, I recommend that you start with Algebraic Expressions, Algebraic Equations, Algebraic Factoring and Exponents. These will give you the foundation needed to understand the other basic algebra topics.
If you just want to find information to solve a problem, go to the topic that is most obviously relevant to your problem, and read. What if you can't solve a problem with the knowledge from one topic? Recognize that some problems might need a combination of different techniques from Pre Algebra and basic Algebra. Identify the techniques and you'll solve the problem easily.
Your topic request will appear on a Web page exactly the way you enter it here. You can wrap a word in square brackets to make it appear bold. For example [my request] would show as my request on the Web page containing your request.
TIP: Since most people scan Web pages, include your best thoughts in your first paragraph.
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Description: Algebra Assistant, the first module of a three-program learning system, allows introductory and intermediate-level students to work through algebra problems in detail, step-by-step.
Reviewer Comments:
The products are cumulative in nature: Calculus Assistant contains all of the materials included in both Pre-Calculus Assistant and Algebra Assistant. Pre-Calculus Assistant contains all of the problems and materials found in the initial offering, Algebra Assistant. As might be expected, Calculus Assistant is the most expensive package ($195--single user; $995--10-user network license).
The features of all three products are identical. Students can quickly check their work on problems by comparing answers with the computer. The disks include more than 6,700 problems; teachers and students can add new problems or change existing problems. All of the work can be printed for student work away from a computer, and it can be saved on a disk to be revisited later.
Algebra Assistant can offer hints so that students see only a single step of a problem, or it can work the entire problem showing each step (complete with annotation). As many as six different graphs can be produced on the screen simultaneously.
The program differs from the more powerful computer algebra systems in that it allows simple step-by-step interaction with the student.
With the straightforward menu, a student may select what he or she believes to be the next best, most plausible, step toward a solution. This simplicity frees teachers from spending additional time teaching the special codes and syntax that computer algebraic manipulators require.
The program's strength is its ability to remove the fear that some students have. It can eliminate trivial mistakes, offer hints, show what operation to perform next, perform the next step automatically, or provide a complete step-by-step solution.
Some learners need to know that the last step they tried was correct before they can move on to the next stop in a problem. For students who fear math, fear being wrong, or fear uncertainty, Algebra Assistant can be a constant reassurance that doesn't add to the teacher's load.
My experience with the 800-phone number was terrible! It took four tries to reach someone who would answer my questions about pricing information. I left voice messages twice and never received a single response.
I did not try Internet support because the product worked the first time, and the included user's guide suggested simple solutions to the "How-to" questions.
Algebra Assistant can be used as an additional method of reaching math students who aren't responding to current approaches for any variety of reasons. It is a simple and inexpensive method to add another way to help students to understand mathematics.
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Mathematics (Specification A) is designed for use in schools and colleges. It is part of a ... appreciate the importance of mathematics in society, employment and study. ...... Edexcel (Oxford University Press, 2007) ISBN 9780199152629 ... Turner D and Potts I – Longman Mathematics for International GCSE Practice Book 2 ...
The International General Certificate of Secondary Education (IGCSE) has been designed ...... The text book that we use is IGCSE Mathematics, second edition Rick .... Physical Education Through Diagrams – GBP 2 – Oxford Revision Guides
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MCP Mathematics: Level A
Book Description: MCP Mathematics promotes mathematical success for all students, especially those who struggle with their core math program. This trusted, targeted program uses a traditional drill and practice format with a predictable, easy-to-use lesson format. MCP Math is flexible and adaptable to fit a variety of intervention settings including after school, summer school, and additional math instruction during the regular school day.By teaching with MCP Math, you can: Provide targeted intervention through a complete alternative program to core math textbooks. Help students learn and retain new concepts and skills with extensive practice. Prepare students at a wide range of ability levels for success on standardized tests of math proficiency
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This work is an introduction to mathematical analysis at an elementary level. Emphasis is given to the construction of national and then real numbers, using the method of equivalence classes and of Cauchy sequences. The text includes the presentation of: sequences of real numbers, infinite numerical series, continuous functions, deriviatives and Ramon-Darboux integration. There are also sections on convex functions and on metric spaces, as well as an elementary appendix on logic, set theory and functions.
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CBSE students will find TARGET CBSE Mathematics (Class - XII) a handy companion, both for revision and preparing for exams.
Summary Of The Book
TARGET CBSE Mathematics (Class - XII) is loaded with useful features that will prove to be a great help to students. Each of the chapters feature a section called 'Take a Look' which summarizes and highlights the important parts of that chapter. There are several exercises with both short and long questions for students to test what they have studied. The book also contains HOTS-based and NCERT questions along with the answers.
The book features questions based on the CBSE-compliant marking pattern. These include one, two, three, and five mark questions. Objective questions like True or False, Reasoning, Fill-in the blanks, and more are also covered in this book. At the end of each chapter, self-assessment tests are included for students to put their knowledge to test and decipher how much they have grasped.
There are also sample question papers along with the solutions, allowing students enough scope to practise before their exams and give themselves mock tests. The board examination papers of previous years are also provided for students to understand how their papers will be set.
Some of the chapters covered in this book include inverse trigonometric functions, three-dimensional geometry, continuity and differentiability, linear programming, relations and functions, area of bounded regions, determinants and matrices, all based on the CBSE syllabus.
TARGET CBSE Mathematics (Class - XII) features a glossary for students to understand important terms from the book better. The book includes question papers from the 2011 and 2012 CBSE Board Examinations and their solutions.
this is one of the bst buk for preparation of board exams.....i and my friends found this buk to be very helpful for our preparation of board exams....
i would like to suggest all of you to buy this book only from flipkart.com....and start preparing for your exams from now onwards with the help of this amazing book....by mc graw hill.........best of luck for your board exams :)
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down to Mathematics: v. 1
Countdown to Mathematics has been written to help self-study students to revise and practise basic skills in arithmetic, algebra, geometry, graphs ...Show synopsisCountdown to Mathematics has been written to help self-study students to revise and practise basic skills in arithmetic, algebra, geometry, graphs and trigonometry. The nine teaching modules in Countdown to Mathematics have been split into two separate books. Volume 1 consists of Modules 1-4 and concentrates on basic mathematical skills. It deals with arithmetic, simple algebra, how to plot and read graphs, and the representation of data. Where possible, the techniques are illustrated with real-world applications. Volume 2 consists of Modules 5-9 and covers geometry, graphs, trigonometry and algebra.. The emphasis here is on the manipulative skills which are necessary for most mathematical courses beyond GCSE standard 256 p. Illustrations
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9780387907ometry: A High School Course
At last: geometry in an exemplary, accessible and attractive form! The authors emphasise both the intellectually stimulating parts of geometry and routine arguments or computations in concrete or classical cases, as well as practical and physical applications. They also show students the fundamental concepts and the difference between important results and minor technical routines. Altogether, the text presents a coherent high school curriculum for the geometry course, naturally backed by numerous examples and exercises.
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Eighth Grade Course Descriptions
810 Algebra
This course will develop the fundamental principals of algebra. Course topics will include algebraic symbolism, simplifying equations, solutions to elementary equations and graphic representations associated with variables. This course will introduce algebraic processes applied to word problems.
820 Accelerated Algebra I
This course, incorporating the consistent use of real numbers and a problem solving approach, emphasizes the principles of algebra, including algebraic symbolism, simplifying complex expressions, solutions to linear and quadratic equations, and graphic representations associated with variables. Students will apply algebraic representations to word problems and analyze the nature of changes in linear and non-linear relationships.
830 Accelerated Geometry
This accelerated course is a comprehensive study of plane and solid geometry including constructions, formulas for measurement and formal proofs. It is based on the axioms and theorems that relate points, lines, planes and solids. Many of the topics are covered in great depth, especially area and volume of solids. Additional emphasis is placed on the integration of algebraic techniques in solving geometric problems.
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Can you suggest me a comprehensive book to revise high school mathematics (up to besic calculus)? It should be extremely clear and complete and "scientific" (not like most high school books). Thank ...
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MATH 104 or equivalent. Completes the study of algebraic and trigonometric skills necessary for successful study of calculus. Trigonometric functions and identities are applied to analytic geometry. Systems of equations and inequalities are solved using algebraic, graphical and matrix/determinant methods. Theory of equations including remainder, factor and De Moivre's theorem are used to study and help in graphing of equations. Introduces series and sequences (arithmetic and geometric), the binomial theorem, and mathematical induction. Assistance is available in the Center for Academic Success. A scientific calculator is required. Three class hours weekly.
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Foundations of Real World Math
"Why is math important? Why do I have to learn math?" These are typical questions that you have most likely asked at one time or another in your education. While you may learn things in math class that you will not use again, the study of mathematics is still an important one for human development. Math is widely-used in daily activities (e.g. shopping, cooking, etc.) and in most careers (e.g. medicine, teaching, engineering, construction, business, statistics in psychology, etc.). Math is also considered a "universal language." One of the fundamental reasons why you learn math is to help you tackle problems, both mathematical and non-mathematical, with clear, concise, and logical steps. In this course, you will study important fundamental math concepts.
This course begins your journey into the "Real World Math" series. These courses are intended not just to help you learn basic algebra and geometry topics, but also to show you how these topics are used in everyday life. In this course, you will cover some of the most basic math applications, like decimals, percents, and even the dreaded "f-word"–fractions. You will not only learn the theory behind these topics, but also how to apply these concepts to your life. You will learn some basic mathematical properties, such as the reflexive property, associative property, and others. The best part is that you most likely already know them, even if you did not know the proper mathematical names.
Let's start with fractions. Have fractions ever been bothersome to you? Do you think that there is no purpose for them? In this course, you will learn that fractions are all around us in the forms of measurement, ratios, and proportions–and we think you might change your tune on the subject. You will see how to solve those sometimes troubling fraction problems, like the ones that use 1 ? and 3 ?, which don't divide as evenly as you'd like. In case you're not yet familiar with fractions, let's offer a common every day example: a recipe for making chocolate chip cookies. You see a recipe that calls for 2 ? cups of flour, ¾ cup of sugar, and ½ teaspoon of vanilla, and you need to make 2 ½ the recipe amount. Each of these measurements involves fractions. If you want to make the right amount of cookies, you have to determine how much you need of each ingredient.
This course will also introduce you to decimals and percentages, which are widely used in money, finances, and measurement. Decimals are all around you, including when you download applications for your smart phone. Say, for example, you've just purchased the newest Angry Birds application for $0.99. The number 0.99 is a decimal. If you want to spend no more than $10.00, then you will have to determine how many other applications you can download without going over budget. In this course, you will learn how to solve complex decimal problems, such as 13.4561 – 21.03 and 301.21 * 140.31.
You will also learn to write ratios and solve proportions in the course. You are probably already very familiar with ratios, even if you're not aware of it. A recipe that calls for "2 parts milk to 1 part flour," or a speed limit sign that reads "55 miles per hour," or a newspaper ad listing apples at a cost of $2.99 per pound — these are all examples of ratios. Ratios and proportions are particularly useful when doing an everyday activity like planning a party: "If I need two hams for nine guests, how many hams will I need for thirty guests?" Learning how to set up and solve problems like this is a very useful mathematical concept that is applicable to real life situations.
Finally, have you wondered how graph and charts are created with certain data? Data can be visually represented in various forms (bar graphs, circle graphs, etc.) to convey information to a reader. In this course, you will see data in common forms and will have to interpret data (for example, reading a chart of the most downloaded songs from iTunes or interpreting football statistics for your fantasy league). The final unit of the course pertains to charts and graphs and includes the interpretation and creation of various charts and graphs.
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. Please pay special attention to Units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units. Throughout the course, there are activities assigned from the Pre-Algebra Textbook that you will need to complete. You will also need to complete:
Subunit 1.4 Assessment
Sub-subunit 2.3.4 Assessment
Sub-subunit 2.4.2 Assessment
Sub-subunit 2.5.2 Assessment
Sub-subunit 3.2.1.5 Assessment
Sub-subunit 3.2.2.4 Assessment
Sub-subunit 3.2.6 Assessment
Sub-subunit 4.2.4 Assessment
Sub-subunit 5.2.2 Assessment
Sub-subunit 6.1.2.2 Assessment
Sub-subunit 6.3.2 Assessment
Subunit 7.2 Assessment
Subunit 7.3 Assessment
Subunit 7.4 Assessment
Subunit 7.5 Assessment
Subunit 7.7 Assessment
The Final Exam
Please note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through the readings, lectures, activities, and assessments listed above as well as all resources in each unit.
In approximately 140.5 hours to complete. Each unit includes a "time advisory" that lists the amount of time you are expected to spend on each subunit. It may be useful to take a look at these time advisories and determine how much time you have over the next few weeks to complete each unit and to then set goals for yourself. For example, Unit 1 should take approximately 9.75 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 4 hours) on Monday night; subunit 1.2 (a total of 3.5 hours) on Tuesday night; subunits 1.3 and 1.4 (a total of 2.25 hours) on Wednesday night; etc.
Tips/Suggestions: Please make sure to take comprehensive notes as you work through each resource. Complete all practice problems, because this will allow you to fully understand each concept. These notes will serve as a useful review as you study for your Final Exam
NOTE: Each of the learning outcomes listed below has been aligned with one or more of the Common Core standards in mathematics. This alignment is reflected in the numbered notation listed alongside each outcome below. For more information on Common Core standards, please read here.
Upon successful completion of this course, the student will be able to:
Apply properties of operations as strategies to add and subtract. (1.OA.B.3)
Apply properties of operations as strategies to multiply and divide. (3.OA.B.5)
Explain how negative numbers are used together to describe quantities having an opposite direction. (6.NS.C.5)
Solve real-world and mathematical problems involving the four operations(including fractions and decimals). (4.MD.A.2)
Find the greatest common factor and least common multiple of whole numbers. (6.NS.B.4)
Just as in life, there are certain things in math that make you shrug and say, "Well, duh. I knew that; it's common sense." This unit will discuss some of the basic algebraic properties which you already know, but may not necessarily know the names of, because they are what some math teachers refer to as the "common sense" properties.
The really neat thing about these properties is that you can see their uses in everyday, non-mathematical ways. For example, if you drive to work, you "commute." Whether you are driving to work from home, or to home from work, you are making the same trip. (Ignoring those times you take a back road because you do not want to spend two hours sitting on the interstate, of course!) In math, the commutative property tells us when we can move numbers around and still get the same answer. Another example is the associative property. The people you hang out with are also known as your "associates." If you are hanging out with two friends, but one of them is in a different room, you still have the same group of friends. The same applies to certain mathematical situations. If you are grouping numbers, depending upon the situation, the grouping is not going to change anything.
Instructions: Please click on the link above and take notes as you watch this video to learn about the Commutative Law of Addition (also known as the Commutative Property of Addition). Watch the presentation carefully two or three times until you are able to explain how changing the order of the addition of two numbers obtains the same result.
Watching this lecture and pausing to take notes should take less than 15 minutes.
Instructions: Please click on the link above and study the "Associative Property of Addition" on pages 15 and 16 of the textbook, stopping at "Grouping Symbols." The material may also be located through the bookmark on the left side (1 The Whole Numbers, 1.2 Adding and Subtracting Whole Numbers, "The Commutative Property of Addition"), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Commutative Property of Addition.
Reading these sections should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and take notes as you watch this video to learn about the Associative Law of Addition (also known as the Associative Property of Addition). Watch the presentation carefully two or three times until you are able to explain how to associate the addition of three numbers to obtain the same result.
Watching this lecture and pausing to take notes should take less than 15 minutes.
Instructions: Please click on the link above and study the "Associative Property of Addition" on page 17 of the textbook, stopping at "The Additive Identity." The material may also be located through the bookmark on the left side (1 The Whole Numbers, 1.2 Adding and Subtracting Whole Numbers, "The Associative Property of Addition"), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Associative Property of Addition adding zero to any number is the original number.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study "The Additive Identity" located on pages 17 and 18 of the textbook, stopping at "Adding Larger Whole Numbers." The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.2 Adding and Subtracting Whole Numbers, "The Additive Identity"), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Additive Identity Property.
Reading this section should take less than 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above; complete the odd-numbered exercises for 11–27 on page 25 and the odd-numbered exercises for 67–79 on page recognize addition laws and properties and apply the conceptsApplications – Geometry" on pages 21 and 22 of the textbook. The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.2 Adding and Subtracting Whole Numbers, "Applications - Geometry"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Example 5 on page 21 and Examples 6 and 7 on page 22. This reading provides applications to the properties and laws of addition 51–65 on pages 26 and apply addition laws and properties "Additive Inverse" located on page 120 of the textbook, including Example 8. The material can also be located through the bookmark on the left side (2 The Integers, 2.2 Adding Integers, "Properties of Addition of Integers"), which will take you directly to the reading. After you study and read this section, complete the "You Try It" problem next to Exercise 8. This reading provides an example of the property and the formal definition of the Additive Inverse Property Inverse Property of Addition results in zero changing the order of the multiplication of two numbers obtains the same result.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above; study the introduction to "Multiplication and Division of Whole Numbers" on page 33 and continue through "The Commutative Property of Multiplication" on page 34. The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.3 Multiplication and Division of Whole Numbers), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Commutative Property of Multiplication.
Reading this section should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and study "The Associative Property of Multiplication" on page 35 of the textbook. The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.3 Multiplication and Division of Whole Numbers, "The Associative Property of Multiplication), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Associative Property of Multiplication.
Reading this section should take less than associate the multiplication of three numbers to obtain the same result multiplying any number by one results in the original number.
Watching this lecture and pausing to take notes should take less than 15 minutes.
Instructions: Please click on the link above and study "The Multiplicative Identity" on page 34 of the textbook. The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.3 Multiplication and Division of Whole Numbers, "The Multiplicative Identity"), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Multiplicative Property 5–27 on page 44 recognize addition laws and properties and apply these concepts. The solutions to these problems are located in the "Answers" section on page 49 section titled "Application –Area" through Example 4 on pages 42 and 43 of the textbook. The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.3 Multiplication and Division of Whole Numbers, "Applications - Area"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Example 4 on page 43. This reading provides applications to the properties and laws of multiplication.
Reading this section should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and complete the odd-numbered exercises for 49–59 on pages 45 and 46 find the area of rectangles as well apply multiplication such as if a math tutor was paid $20 per hour and worked 20 hours, how much would the tutor get paid? inverse property of multiplication results in one.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study "The Multiplicative Inverse Property" through Example 1 on pages 266 and 267 of the textbook. The material can also be located through the bookmark on the left side (4 Fractions, 4.3 Reciprocals, "The Multiplicative Inverse Property"), which will take you directly to the material. After you read and study this section, attempt the "You Try It" problem beside Example 1. Check your answer on page 267. This reading provides an example of the property and the formal definition of the Multiplicative Inverse Property.
Reading this section should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Link: College of the Redwoods: Department of Mathematics' Pre-Algebra Textbook, 2nd Edition: "Multiplication by Zero" (PDF)
Instructions: Please click on the link above and study the section titled "Multiplication by Zero" on pages 34 and 35 of the textbook. You may stop when you reach the section titled "The Associative Property of Multiplication." The material can also be located through the bookmark on the left side (1 Whole Numbers, 1.3 Multiplication and Division of Whole Numbers, "Multiplication by Zero"), which will take you directly to the reading. This reading provides an example of the property and the formal definition of the Multiplication by Zero.
Reading this section should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and study the section titled "Division by Zero Is Undefined" on page 40 of the textbook, stopping at "Dividing Larger Whole Numbers." The material can also be located through the bookmark on the left side (1 The Whole Numbers, 1.3 Multiplication and Division of Whole Numbers, "Division by Zero is Undefined"), which will take you directly to the reading. This reading provides an example of the property and the formal definition of why division by zero is undefined 69–75 on page 46 of the textbook. The apply multiplication laws and properties. why dividing by zero is undefined.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the links above and take notes as you watch these videos. Watch the first presentation carefully two or three times until you are able to explain how the distributive property applies with an addition expression. Watch the second presentation carefully two or three times until you are able to explain how the distributive property applies with a subtraction expression.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read "The Distributive Property" section through "Sample Set A." Then, complete "Practice Set A," exercises 1-7. The solutions to the problems are revealed below each problem. These problems will allow you to practice using the distributive property to simplify algebraic expressions "Number Properties Assessment" multiple-choice assignment. These assignments incorporate concepts from the Associative, Commutative, Identity, Inverse, and Distributive Properties. You may want to review these concepts associated with the questions by revisiting the Khan videos in this unit.
Completing this assessment should take approximately 30 minutes.
Note: You must be logged into your Saylor Foundation School account in order to access this assessment. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
Unit 2: Order of Operations
In life, we often have procedures that everybody uses to avoid problems. When driving a car, for example: if you want to change lanes, you have to first look to make sure the lane is clear, activate your turn signal, check the lane again, move into the lane, and deactivate your turn signal. You do not move into the lane, activate your signal, make sure the lane is clear, and deactivate your signal. That can, and eventually will, cause a serious accident. In order to avoid costly errors, mathematicians had to agree on the series of steps that are needed to simplify expressions involving the four basic operations, grouping symbols, and exponents. This series of steps is known as the "order of operations" and is more commonly known as either PEMDAS or "Please Excuse My Dear Aunt Sally, she Left to Right." This tells us in which order to simplify the expression. (Tip: it is multiply OR divide and add OR subtract– whichever you see first.)
Mathematicians also needed a way to quickly write out a repeated multiplication problem, like 2 x 2 x 2 x 2 x 2, so they invented the use of exponents. This unit will introduce you to the process of working with basic exponents. As you go higher, you will learn more about exponents.
Another topic you will learn about in this unit is the concept of "greatest common factor." Mathematically, the greatest common factor (GCF) is the largest number you can divide two or more numbers by. In real life, it also makes appearances, both mathematical and non-mathematical. A detective trying to make connections between an arrested criminal and a suspected accomplice is going to be less interested in the facts that they have both eaten at McDonald's and both like strawberry milkshakes than in the fact that the suspected accomplice has been the criminal's best friend for twenty years. That fact is far greater to the investigation.
The last topic you will cover is related to greatest common factor but is different. It is known as "least common multiple." Here, you are trying to determine the smallest number that two numbers can both divide into. Again, it appears in life. Let's say your favorite radio station is running a promotion: every fifth caller receives free concert tickets, and every twelfth caller receives a free gas card. How long will it take before they have a caller who receives both prizes on the same phone call? This is an example of using the least common multiple. (In case you are wondering, it would be the 60th caller who won both prizes.)
Instructions: Please click on the link above and take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to solve problems finding the greatest common divisor/factor.
Watching this lecture and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read the entire section on "The Greatest Common Factor (GCF)" through "Sample Set A." Look closely at the section titled "A Method for Determining the Greatest Common Factor." Then, complete "Practice Set A," exercises 1–4 and the even-numbered problems for 6–20. The solutions to the problems are shown below each problem. These problems will allow you to practice finding the greatest common factor between numbers.
Reading this section and completing this activity should take approximately 1 hour.
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Instructions: Please click on the link above and complete the "Greatest Common Divisor" assignment. This assignment incorporates concepts from the greatest common divisor solve problems finding the least common multiple.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and read the "Least Common Multiple" section. Start with the "Multiples" section, and then complete "Practice Set A," exercises 1–3. Continue with the "Common Multiples" section, and then complete "Practice Set B," exercises 6–8. Finish by reading "The Least Common Multiple (LCM)" section, and then complete "Practice Set C," exercises 11–13. Then, complete the even-numbered problems for 16–44. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice finding the least common multiple between numbers.
Reading this section and completing this activity should take approximately 2 hours.
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Instructions: Please click on the link above and complete the "Least Common Multiple" assignment. This assignment incorporates concepts from the least common multiple Click on the link above and watch this video, which consists of two least common multiple and greatest common factor word problems. Pause the video after each problem has been given, and try to work out the answer on your own before coming back to the video to check your answer. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2.
Instructions: Please click on the link above and study the section titled "Adding Integers with Like Signs" on pages 115–117 of the textbook. You may stop when you reach the section titled "Adding Integers with Unlike Signs" on page 117 Examples 1 and 2 on page 115 and Example 3 on page 116. This material provides examples of adding two positive and two negative integers add integers with different signs.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study the section titled "Adding Integers with Unlike Signs" on pages 117–119 of the textbook. You may stop when you reach the section titled "Properties of Addition of Integers" on page 119 Example 4 on page 117 and Example 5 on page 118. This material provides examples of adding one positive and one negative integer35 and 65–83 on pages 124 and 125 of the textbook. The exercises can also be located through the bookmark on the left side (2 The Integers, 2.2 Adding Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to recognize addition properties as well as determine the profit and loss for a company. The solutions to these problems are located in the "Answers" section on page 126 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.2 AddingSubtracting Integers" on pages 128 through 132. The material can also be located through the bookmark on the left side (2 The Integers, 2.3 Subtracting Integers, "Subtracting Integers"), which will take you directly to the reading. After reading and studying the "Subtracting Integers" section, complete the "You Try It" problems beside Example 1 on page 129 and Example 4 on page 131. This material provides examples of subtracting integers and 51–59 on pages 133 and 134 of the textbook. The exercises can also be located through the bookmark on the left side (2 The Integers, 2.3 Subtracting Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to recognize subtraction and properties and apply the concepts when dealing with a temperature change and comparing highest and lowest points. The solutions to these problems are located in the "Answers" section on page 135 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.3 Subtracting "Adding Negative Numbers" assessment. This quiz quiz, you will compute your answer and type it into the answer box. You may then click on "Check Answer" to see if you were correct or if you need to try again.
Instructions: Please click on the link above and complete the "Negative Number Word Problems" assignment. This assignment multiply integers with different signs.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study the sections titled "Multiplication and Division of Integers" on pages 137 and 138 as well as "Multiplying by Minus One" and "The Product of Two Integers" on pages 140–142 of the textbook. The material can also be located through the bookmark on the left side (2 The Integers, 2.4 "Multiplication and Division of Integers") and (2 The Integers, 2.4 Multiplication and Division of Integers, "Multiplying by Minus One"), which will take you directly to the readings. After reading and studying these sections, complete the "You Try It" problems beside Examples 1 and 2 on pages 141 and 142. This material provides examples of multiplication of integers and rules associated with each problem 17–47 as well as problem 85 on pages 145 through 147 multiplication divide integers with different signs.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study the section titled "Division of Integers" on pages 143 and 144 of the textbook. The material can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Division of Integers"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Example 4 on page 144. This reading provides examples of division of integers and rules associated with each problem.
Reading this section and completing the exercises should take approximately 20 minutes.
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Instructions: Please click on the link above and complete the odd-numbered exercises for 61–83 division "Multiplying and Dividing Negative Numbers" assignment. This assignment incorporates concepts from multiplying and dividing explain how to write a problem in exponential notation. Watch the second presentation carefully two or three times until you are able to explain how to simplify a problem in exponential notation.
Watching these lectures and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and read the entire section titled "Exponential Notation." Read the introductory text on exponential notation through "Sample Set A." Then, complete "Practice Set A," exercises 1–6. Continue with the "Reading Exponential Notation" subsection. Then, complete the odd-numbered exercises for 15–29 and 31–57. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to understand the basics of exponential notation.
Reading this section and completing this activity should take approximately 1 hour and simplify exponents with a positive and negative bases as well as exponents to the zero power.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the link above and study Example 3 located on page 142 of the textbook. After reading and studying this example, complete the "You Try It" problems beside Example 3. This material provides an example of exponential notation with negative signs and bases.
Studying this example and attempting the "You Try It" example should take approximately 15 minutes.
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Instructions: Please click on the link above and complete the odd-numbered problems for 49–59 simplify exponent problems with negative bases complete the assessment that tests your knowledge on positive and zero exponents. This incorporates concepts from positive exponents with a positive or negative base and zero exponents. You can review the concepts associated with the questions with the Khan videos in Unit 2. Answer each question by inputting your calculation into the answer box. You may click on "Check Answer" to check if your answers are correct or if you need to try again.
Instructions: Please click on the link above and take notes as you watch this video. Watch the presentation carefully two or three times until you are able to apply the process of the order of operations.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the link above and read the section titled "Order of Operations with Integers" on pages 148–150 of the textbook, stopping at "Evaluating Fractions." The material can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 2 and 4 on page 149. This material provides examples of order of operations with integers. Note that this reading also covers the topic outlined in sub-subunit 2.5.2 problems for 1–39 on page 152 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to apply the process of the order of operations The topic of this sub-subunit is covered by the reading assigned below sub-subunit 2.5.1. To apply your knowledge of this topic, also complete the "You Try It" problem beside Example 3 on page 149.
Completing this activity should take approximately 15 minutes.
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Instructions: Please click on the link above and complete the odd-numbered problems for 45–53 as well as 59, 61, 67, 69, 73, 75, and 77 on page 153 45 minutes.
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Instructions: Please click on the link above and review the section titled "Evaluating Fractions," including Example 6, on page 150 of the textbook. The material can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Evaluating Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problem beside Example 6 on page 150. This material provides examples of order of operations while evaluating fractions problems for 81–103 on pages 153 and 154, which tests your knowledge on order of operations. You can review these concepts associated with the questions with the Khan videos in subunit 2.5. Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again.
The very word – fractions – often chills the bones of math students. Working with fractions is easily the most dreaded, most feared topic in any math class. However, fractions are actually very easy to work with, if you learn the rules. After all, you cannot escape fractions in life; they are everywhere.
Have you ever eaten a Hershey's chocolate bar? It is conveniently broken up into little pieces, allowing you the option to devour in big bites or to savor tiny little morsels. Let's say you have a Hershey's bar sitting on your dining room table. Your oldest child cheerfully announces that she has eaten half of the bar, and her younger brother has eaten a quarter of the bar. If you know how to work with fractions, you can quickly calculate how much of the bar is left.
Fractions appear in many other situations such as sale prices, measurements, money, gardening; the list of applications is virtually endless. In this unit, you will learn to work with fractions. You will learn how to reduce them, how to add/subtract/multiply/divide them, and how to apply them to real-world situations. One suggestion: never show fear. Fractions can smell fear.
Instructions: Please click on the links above and take notes as you watch these videos (3 minutes each). Watch the first presentation carefully two or three times until you are able to identify the numerator and denominator of a fraction. Watch the second presentation carefully two or three times until you are able to identify parts of a fraction.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read "Fractions of Whole Numbers," "The Parts of a Fraction," and "Reading and Writing Fractions" sections. Then, complete "Practice Set A," exercises 1–5 and "Practice Set B," exercises 6–17. The solutions to the problems are shown directly below each problem. These problems will allow you to practice identifying numerators and denominators as well as writing fractions by using words every fourth problem for 18–46 and the even-numbered problems for 48-64. The solutions to the problems are shown directly below each problem. These problems will allow you to practice identifying numerators and denominators. This includes determining the numerator and denominator from problems such as "you need ¾ of a cup of sugar to make a batch of cookies."
Completing these exercises should take approximately 30 identify equivalent fractions. Watch the second presentation carefully two or three times until you are able to compare fractions.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read the "Equivalent Fractions" section through "Sample Set A." Then, complete "Practice Set A," exercises 1–5. The solutions to the problems are shown directly below each problem. These problems will allow you to practice identifying if pairs of fractions are equivalent assignments, which test your knowledge on equivalent fractions. You can review the concepts associated with the questions the Khan videos in subunit 3 odd-numbered problems 11–25. The solutions to the problems are shown directly below each problem. These problems will allow you to determine a proper fraction, improper fraction, or a mixed number.
Completing these exercises mixed numbers and improper fractions.
Watching this lecture and pausing to take notes should take approximately 30 minutes 15 minutes.
Instructions: Please click on the link above and review the section titled "Changing Improper Fractions to Mixed Fractions" on pages 293–295 of the textbook, stopping at "Multiplying and Dividing Mixed Fractions." The material can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Changing Improper Fractions to Mixed Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 4–6 on pages 294 and 295. This material provides examples of converting an improper fraction to a mixed fraction.
Reading this section and completing the exercises should take approximately 30 minutes.
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Instructions: Please click on the link above and read the "Converting Improper Fractions to Mixed Numbers" section. Then, complete "Practice Set A," exercises 1–6. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting an improper fraction to a mixed numbernumber exercises 27–39. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting an improper fraction to a mixed number.
Completing these exercises should take approximately 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above 30 minutes 41–55. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting a mixed number and an improper fraction assignment that tests your knowledge of converting fractions. This assignment incorporates concepts from converting mixed numbers to improper fractions and vice versa. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.1.3Reducing Fractions to Lowest Term" section. Then, complete "Practice Set B," exercises 6–11 and "Practice Set C," exercises 12–17. The solutions to the problems are shown directly below each problem. These problems will allow you to practice reducing fractions to their lowest terms problems for 61–77 as well as the odd-numbered problems for 89–113. The solutions to the problems are shown directly below each problem. These problems will allow you to practice reducing fractions to their lowest terms as well as determine fractional parts of a day solve an application of raising fractions to highest terms.
Watching this lecture and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read the "Raising Fractions to Higher Terms" section. Then, complete "Practice Set D," exercises 18–22. The solutions to the problems are shown directly below each problem. These problems will allow you to find the missing numerator and denominator 39–53. The solutions to the problems are shown directly below each problem. These problems will allow you to practice reducing fractions to their lowest terms rewrite fractions with a least common denominator.
Watching this lecture and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read the "Addition of Fractions with Like Denominators" section. Then, complete "Practice Set A," exercises 1–4. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice adding fraction fn adding fractions. This assignment incorporates concepts from addingSubtraction of Fractions with Like Denominators" section. Then, complete "Practice Set B," exercises 5–9. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice subtracting fractions with like1. Please click on the link above and complete the odd-numbered problems for 11–35 and 39. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice adding and subtracting fractions and read the "Addition and Subtraction of Fractions" section stopping at "Sample Set A," Example 3. Then, complete "Practice Set A," exercise 1. The solutions to the problem can be revealed by the "Show Solution" link under each problem. This reading and exercise will allow you to practice adding fractions with unlike denominators assignment that tests your knowledge of adding fractions. This assignment incorporates concepts from adding starting at "Sample Set A," Example 3, read through the "Addition and Subtraction of Fractions" section. Then, complete "Practice Set A," exercises 2–5. The solutions to the problems are shown directly below each problem. These problems will allow you to practice subtracting fractions with unlike3. Please click on the link above and complete the even-numbered problems 6–36. The solutions to the problems are shown directly below each problem. These problems will allow you to add and subtract fractions with unlike denominators Fractions with Different Denominators" on pages 277–279 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.4 Adding and Subtracting Fractions, "Adding and Subtracting Fractions with Different Denominators"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 4 and 5 on page 279. This material provides examples of subtracting fractions with different signs103 on pages 287 through 288 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.4 Adding and Subtracting Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to add and subtract fractions with different signs. The solutions to these problems are located in the "Answers" section on page 289 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.4 Adding and Subtracting Fractions that tests your knowledge on adding and subtracting fractions. This quiz incorporates concepts from adding and subtracting fractions with like and View each presentation carefully two or three times until you are able add mixed numbers with common denominators.
Watching these lectures and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and take notes as you watch this video. Watch the presentation carefully two or three times until you are able subtract mixed numbers with common denominators.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the links above and take notes as you watch these videos. View each presentation carefully two or three times until you are able add mixed numbers with unlike denominators.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and study the "Adding Mixed Fractions" section on pages 301–303 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.6 Adding and Subtracting Mixed Fractions, "Adding Mixed Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1, 2, and 3 on pages 301 and 302. This material provides examples of subtracting fractions with different signs.
Reading this section and completing the exercises should take approximately 1 hour.
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Instructions: Please click on the links above and take notes as you watch these videos. View each presentation carefully two or three times until you are able subtract mixed numbers with unlike denominators.
Watching these lectures and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the link above and study the "Subtracting Mixed Fractions" section 9 on pages 304–307 of the textbook, stopping at Example 9. The material may also be located through the bookmark on the left side (4 Fractions, 4.6 Adding and Subtracting Mixed Fractions, "Subtracting Mixed Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems next to Examples 5 through 8 on pages 304–306. This material provides examples of subtracting mixed fractions.
Reading this section and completing the exercises should take approximately 45 minutes.
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Instructions: This topic is covered by the reading and activity in sub-subunits 3.2.2.1, 3.2.2.2, and 3.2.2.3. Please click on the link above and complete the even-numbered problems for 6–36. The solutions to the problems are shown directly below each problem. These problems will allow you to practice with the addition and subtraction of mixed assignments that tests your knowledge of adding and subtracting mixed numbers. These assignments incorporate concepts from adding and subtracting mixed numbers with like and unlike denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.2. Compute the answer to the given problem, and input your response View each presentation carefully two or three times until you are able understand applications where you add and subtract.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and review the "Adding and Subtracting Mixed Fractions" section through Example 9 on pages 301–307 of the textbook; you read this section in sub-subunits 3.2.1 and 3.2.2. The material may also be located through the bookmark on the left side (4 Fractions, 4.6 Adding and Subtracting Mixed Fractions), which will take you directly to the reading. After reviewing this section, complete the "You Try It" problem next to Example 4 on page 303 and Example 9 on page 307. This material contains applications of adding and subtracting fractions.
Reading this section and completing the exercises should take approximately 15 minutes.
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Instructions: Please click on the links above and complete the following problems: problem 39 for "Addition and Subtraction of Fractions with Like Denominators;" problems 50, 52, and 54 for "Addition and Subtraction of Fractions with Unlike Denominators;" problems 42, 44, and 46 for "Addition and Subtraction of Mixed Numbers." The solutions to the problems can be revealed by the "Show Solution" link below each problem, or may be found directly below the problem. These problems will allow you to practice adding and subtracting fractions with unlike denominators, which include finding amounts needed for recipes and finding the cost after an increase.
Reading this section and completing these exercises should take approximately 30 minutes.
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Instructions: Please click on the link above and study the "Multiply Fractions" section on pages 249–252, stopping at Example 5 on page 252 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.2 Multiplying Fractions), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems next to Examples 2, 3, and 4 on pages 251 and 252. This material contains examples of multiplying 5–27 on page 260 multiply fractionsMultiplication of Mixed Numbers" section stopping at the "Powers and Roots of Fraction" section. Then, complete "Practice Set C," exercises 17–20. The solutions to the problems are shown directly below each problem. These problems will allow you to practice multiplying 25–47 on pages 297 and 298 multiplyDivision" section on pages 267 and 268 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Division"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems next to Examples 2, 3, and 4 on pages 267 and 268. This material contains examples of dividing 33–67 on page 271 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to divide fractions. The solutions to these problems are located in the "Answers" section on page 273 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Examples 7–9 under "Division of Fractions" section, stopping at the "Powers and Roots of Fraction" section. Then, complete "Practice Set B," exercises 11–13. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice dividing 49–71 on pages 298 and 299 divide apply the multiplication of fractions to a word problem. Watch the second presentation carefully two or three times until you are able to apply the division of fractions to a word problem.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and study the "Parallelograms" and "Triangles" sections on pages 255–258 of the textbook, stopping at "Identifying the Base and Altitude." The material may also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Parallelograms"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 8 and 9 on pages 256 and 257. This material contains examples of dividing fractions 57–69 on page 262 apply multiplying fractions by finding the areas of parallelograms and triangles problems 73 and 75 on page 299 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Exercises"), which will take you directly to the material. These exercises will provide you with the opportunity to practice dividing fractions by dividing fields and cutting jewelry into pieces assessments that test your knowledge on applications of multiplying and dividing fractions. You can review the concepts associated with these questions with the Khan videos in sub-subunits 3.2.4 and 3.2 read "The Order of Operations" section, stopping at "Sample Set A," Example 5. Then, complete "Practice Set A," exercises 1–5 and 7. The solutions to the problems are shown directly below each problem. These problems will allow you to practice the order of operations with 17–35 on pages 321 and 322 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.7 Order of Operations with Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to practice dividing fractions. The solutions to these problems are located in the "Answers" section on page 324 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.7 Order of Operations withSimple Fractions and Complex Fractions" and "Converting Complex Fractions to Simple Fractions" sections, stopping at "Sample Set A," Example 5. Then, complete "Practice Set A," exercises 1–6. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice the order of operations with complexnumber problems for 7–25. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to use order of operations to simplify complex fractions.
Reading this section and completing the exercises should take approximately 45 minutes.
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Unit 4: Decimals
Congratulations on surviving fractions! In this unit, you will turn your attention to the "fraternal twin" of fractions: decimals. Yes, decimals are really just fractions in disguise! Who knew? For example, look at (American) money. A dollar is 100 cents; a quarter is 25 cents, or in decimal form, $0.25. The fraction 25/100 reduces to ¼, which is read as "one-quarter." Decimals are fractions, and fractions are decimals. It's all in how you write them.
Decimals are everywhere, just like fractions. You cannot go shopping without encountering decimals. Whether you are adding up totals on your shopping list, calculating your change, or even just measuring the length of something, you will use decimals. If you add up all your purchases, find that your total comes to $17.31, and you hand the cashier $20, you need to know how to determine your change to make sure the cashier gives you back the correct amount of money. If you are measuring the length of your wall in order to fit a couch there, you might find that the wall's length is in between two lengths, measuring at, say, 11.5 ft. You have to know how to deal with decimals to approximate distances.
In this unit, you will learn how to add/subtract/multiply/divide decimals as well as how to convert between fraction and decimal form.
Instructions: Please click on the links above and take notes as you watch these videos. Watch the first presentation carefully two or three times until you are able to understand place value. Watch the second presentation carefully two or three times until you are able to write decimals in word form.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and study the "Introduction to Decimals," "Decimal Notation," and "Pronouncing Decimal Numbers" sections on pages 342–346 of the textbook, stopping at "Decimals to Fractions." The material may also be located through the bookmark on the left side (5 Decimals, 5.1 Introduction to Decimals), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–6 on pages 344–346. This material contains examples of decimal notation and pronouncing decimal numbers.
Reading these sections39 on pages 353 and 354 writing out decimals numbers in expanded form and words. The solutions to these problems are located in the "Answers read the "Rounding Decimal Numbers" section through "Sample Set A." Then, complete "Practice Set A," exercises 1–7. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to round decimals to various positions even-numbered problems for 8–22. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will help you practice rounding decimals to various positions.
Reading this section and completing the exercises should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and study the "Comparing Decimals" section on pages 350–352 of the textbook. The material may also be located through the bookmark on the left side (5 Decimals, 5.1 Introduction to Decimals, "Comparing Decimals"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 11 and 12 on pages 351 and 352. This material contains examples of comparing 81–91 on page 355 write out decimals numbers in expanded form and words. The solutions to these problems are located in the "Answers read the "Converting a Fraction to a Decimal" section, stopping at "Sample Set A," Example 5. Then, complete "Practice Set A," exercises 1–4. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to convert fractions into decimals 7–31 and 37–53. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to convert fractions into decimals assignment that tests your knowledge onf converting fractions to decimals. You can review the concepts associated with the questions with the Khan videos in sub-subunit 4.1.4Converting an Ordinary Decimal to a Fraction" section through "Sample Set A," and complete "Practice Set A," exercises 1–4 9–27 assignments that test your knowledge of converting decimals to fractions. You can review the concepts associated with the questions with the Khan videos in sub-subunit 4.1.4Instructions: Please click on the links above and take notes as you watch these videos. View these presentations carefully two or three times until you are able to convert repeating decimals to fractions.
Watching these lectures and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the link above and study the "Adding Decimals" section on pages 359–361 of the textbook, stopping at "Subtracting Decimals." The material may also be located through the bookmark on the left side (5 Decimals, 5.2 Adding and Subtracting Decimals, "Adding Decimals"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–4 on pages 359–361. This material contains examples of adding decimal numbers11 adding decimal assignments that test your knowledge of adding decimals. You can review the concepts associated with the questions with the Khan videos in sub-subunit 4.2.1 subtract decimals. Watch the second presentation carefully two or three times until you are able to add and subtract decimals in application.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study the "Subtracting Decimals" section on pages 361 and 362 of the textbook, stopping at "Adding and Subtracting Signed Decimal Numbers." The material may also be located through the bookmark on the left side (5 Decimals, 5.2 Adding and Subtracting Decimals, "Subtracting Decimals"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 5 and 6 on pages 361 and 362. This material contains examples of subtracting23 and 81–87 apply the addition and subtraction of decimals assignments that test your knowledge of subtracting decimals. You can review the concepts associated with the questions with the Khan videos in sub-subunit 4.2 Signed Decimal Numbers" section on pages 362–364 of the textbook. The material may also be located through the bookmark on the left side (5 Decimals, 5.2 Adding and Subtracting Decimals, "Adding and Subtracting Signed Decimal Numbers"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 7–10 on pages 362–364. This material contains examples of adding and subtracting signed 25–63 add and subtract signed decimalsMultiplying Decimals" section on pages 370–373 of the textbook, stopping at "Multiplying Signed Decimal Numbers." Then, read "The Circle" section on pages 376–380. The material may also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying Decimals) and (5 Decimals, 5.3 Multiplying Decimals, "The Circle"), which will take you directly to the readings. After reading and studying these sections, complete the "You Try It" problems beside Examples 1–3 on pages 370–373 and Examples 9 and 10 on pages 378–380. This material contains examples of multiplying decimal numbers and its applications-27 and 89-105 decimals numbers and study applications of multiplying decimals, which include finding the total cost of items. The solutions to these problems are located in the "Answers" section on page 384 of the textbook. The solutions page can also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying Decimals the "Multiplying Signed Decimal Numbers" section on pages 373 and 374 of the textbook, stopping at "Order of Operations." The material may also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying Decimals, "Multiplying Signed Decimal Numbers"), which will take you directly to the material. After reading and studying this section, complete the "You Try It" problems beside Examples 4 and 5 on pages 373 and 374. This reading contains examples of multiplying signed 29–55 signed decimals numbers. The solutions to these problems are located in the "Answers" section on page 385 of the textbook. The solutions page can also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying multiplying decimals. You can review the concepts associated with the questions with the Khan videos in sub-subunit 4.2.2.1, and 4.2.2Dividing Decimals" on pages 386–390 of the textbook, stopping at "Dividing Signed Decimal Numbers." The material may also be located through the bookmark on the left side (5 Decimals, 5.4 Dividing Decimals), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–3 on pages 387–390. This reading contains examples and applications of dividing39 and 107–113 on page 395 of the textbook. These exercises can also be located through the bookmark on the left side (5 Decimals, 5.4 Dividing Decimals, "Exercises"), which will take you directly to the material. These exercises will provide you with the opportunity to divide decimals numbers and study applications of dividing decimals, which include finding averages for a project "Dividing Signed Decimal Numbers" section on pages 390 and 391 of the textbook, stopping at "Rounding." The material may also be located through the bookmark on the left side (5 Decimals, 5.4 Dividing Decimals, "Dividing Signed Decimal Numbers"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Example 4 on page 390. This reading contains examples of dividing signed decimal numbers exercises for 41–63 on page 395 of the textbook. These exercises can also be located through the bookmark on the left side (5 Decimals, 5.4 Dividing Decimals, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to divide signed decimals numbers dividing decimals. You can review the concepts associated with the questions with the Khan videos in sub-subunit 4.2.3.1, and 4.2.3.2. Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again.
Link: College of the Redwoods: Department of Mathematics' Pre-Algebra Textbook, 2nd Edition: "Order of Operations" (PDF)
Instructions: Please click on the link above and study the "Order of Operations" section on pages 374 and 375 of the textbook, stopping at "Powers of Ten." Then, read "Order of Operations" through Example 8 on pages 393 and 394. The material may also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying Decimals, "Order of Operations")and (5 Decimals, 5.4 Dividing Decimals, "Order of Operations"), which will take you directly to the readings. Review only the processes involved with each problem and do not worry about substituting into the expression. This material contains examples of order of operations79 on page 382 and 89–99 on page 397 of the textbook. These exercises can also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying Decimals, "Exercises") and (5 Decimals, 5.4 Dividing Decimals, "Exercises"), which will take you directly to the assignments. These exercises will provide you with the opportunity to apply order of operations to decimal numbers. The solutions to these problems are located in the "Answers" section on page 384 and page 399 of the textbook. The solutions page can also be located through the bookmark on the left side (5 Decimals, 5.3 Multiplying Decimals, "Answers") andUnit 5: Ratios and Proportions
In this unit, you will study ratios and proportions. These are mathematical concepts you use all the time, probably without even realizing it. Have you ever been in line at a donut store, comparing the number of chocolate donuts to the number of customers? That's a ratio. Perhaps you are telling your vet how many times a week your dog drags you outside for an extended walk. That's also a ratio. Have you ever been driving on a trip, going around 75 mph, and wanted to know how long it would take to reach your destination, which was only 35 miles away? You would find the answer using a proportion. In sports, statisticians use proportions to predict an athlete's production, based on what they've done up to that point. In this unit, you will learn how to write ratios, how to set-up and solve proportions, and how to apply these skills to real-world experiences.
Instructions: Please click on the link above and study the "Introduction to Ratios and Rates" section on pages 449–451 of the textbook, stopping at "Rates." The material may also be located through the bookmark on the left side (6 Ratio and Proportion, 6.1 Introduction to Ratios and Rates), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1 and 2 on pages 449–451. This material contains examples on ratios write and simplify ratios these videos. Watch the first presentation carefully two or three times until you are able to find unit rates. Watch the second presentation carefully two or three times until you are able to find unit prices.
Watching these lectures and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study the "Rates" and "Unit Rates" sections on pages 451–453. The material may also be located through the bookmark on the left side (6 Ratio and Proportion, 6.1 Introduction to Ratios and Rates, "Rates"), which will take you directly to the reading. After reading and studying these sections, complete the "You Try It" problems beside Examples 3–6 on pages 451–453. This material contains examples on rates and unit rates 23–37 write and simplify rates, which includes comparing rates to see the better deal on an item assignment that tests your knowledge of ratio word problems. This assignment incorporates concepts of ratios. You can review the concepts associated with the questions with the Khan video in sub-subunit 5Introduction to Ratios and Rates" section on pages 456–459 of the textbook, stopping at "Example 4." The material may also be located through the bookmark on the left side (6 Ratio and Proportion, 6.2 Introduction to Proportions), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–3 on pages 457–459. This material contains examples on an introduction to proportions13, and 17, 19, 23, 27, 29, 31, and 35 on page 463 solve proportions review the "Introduction to Ratios and Rates" section through Example 4 on pages 456–459 of the textbook; note that you already studied this material in sub-subunit 5.2.1. The material may also be located through the bookmark on the left side (6 Ratio and Proportion, 6.2 Introduction to Proportions), which will take you directly to the reading. After reviewing this section, complete the "You Try It" problems beside Example 4 on page 459. This material contains examples on an introduction to proportions 37-53 on pages 464 and 465 write and solve proportions, which include finding the cost of an item the "Applications of a Proportion" section. Then, complete "Practice Set A," exercises 1–5. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice applications of proportions odd-numbered problems for 7–25. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice applications of proportions.
Reading this section and completing the exercises should take approximately 45 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the links above and complete the assignments that test your knowledge on proportion word problems. These assignments incorporate concepts of ratios and proportions. You can review the concepts associated with the questions with the Khan video in sub-subunit 5.2In this course, you have already studied fractions and decimals. In this unit, you will study the other "fraternal twin" of fractions: percents, which are actually fractions and decimals in disguise. (Perhaps we should call them "fraternal triplets.") Going back to our example with decimals: we established that a dollar is 100 cents, a quarter is 25 cents, and the fraction form would be 25/100, which reduces to ¼. A percentage is simply a fraction whose denominator is 100. Therefore, 25/100 becomes 25%. Because it is also 0.25, the percent is a fraction which is a decimal, which in turn is a percent. It's the Circle of Math. (Cue music from "The Lion King.")
Percents appear all over the place in life, especially when it comes to buying products. If you are considering whether to buy clothes at one store that has a sale with 65% off or a second store that has a sale with 50% off and an additional 15% discount off the sale price, you might be surprised to learn that the two sales are not the same. For those who follow the stock market, you might see the news talking about how your stock has had an increase of 70%. What does that mean?
In this unit, you will learn the rules of percentages and how to apply them. You will learn to convert percentages to and from fractions and decimals. You will learn about percent increase and decrease, which comes into play when you are out shopping. You will also learn (to the delight of shoppers everywhere) exactly how to calculate sale prices, restaurant tips, and other similar items.
Instructions: Please click on the link above and take notes as you watch this video. Watch this presentation carefully two or three times until you are able to understand how to convert a decimal to a percent and a percent to a decimal.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the links above and study the "Changing a Percent to a Decimal" and "Changing a Decimal to a Percent" sections on pages 504 and 505 of the textbook. The material may also be located through the bookmark on the left side (7 Percent, 7.1 Percent, Decimal, Fractions, "Changing a Percent to a Decimal), which will take you directly to the reading. After reading and studying these sections, complete the "You Try It" problems beside Examples 4–7 on pages 504 and 505. This reading contains examples on changing a percent to a decimal and vice versa.
Reading this section should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and complete the odd-numbered exercises for 19–49 on pages 508 and 509 video. View these presentations carefully two or three times until you are able to understand how to convert a fraction to a percent and a percent to a fraction.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the links above and study the "Changing a Percent to a Fraction" section on pages 502–504 of the textbook, stopping at "Changing a Percent to a Decimal." Then, study "Changing a Fraction to a Percent" on pages 506–508 of the textbook. The material may also be located through the bookmark on the left side (7 Percent, 7.1 Percent, Decimal, Fractions, "Changing a Percent to a Fraction"), and (7 Percent, 7.1 Percent, Decimal, Fractions, "Changing a Fraction to a Percent"), which will take you directly to the readings. After reading and studying these sections, complete the "You Try It" problems beside Examples 1–4 on pages 502–504 and Examples 8–10 on pages 506–508. This reading contains examples of changing a percent to a fraction and vice versa.
Reading this section and completing the exercises should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the links above and complete the odd-numbered exercises for 1–17 and 51–79 on pages 508–510 "Find a Given Percent of a Given Number" on pages 512–514 of the textbook, stopping at "Find a Percent Given Two Numbers." The material may also be located through the bookmark on the left side (7 Percent, 7.1 Percent, Decimal, Fractions, "Find a Given Percent of a Given Number), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–3 on pages 512–514. This reading contains examples of finding a given percent of a given a percent given two numbers.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study "Find a Percent Given Two Numbers" on pages 514–516 of the textbook, stopping at "Find a Number That Is a Given Percent of Another Number." 4 and 5 on pages 514–516. This reading contains examples of finding a percent when given two numbers.
Reading this section should take approximately 30 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and study "Find a Number That Is a Given Percent of Another Number" on pages 516 and 517 of the textbook. 6 and 7 on pages 516 and 517. This material contains examples of finding a number that is a given percent of another 1–49 on page 518 519 of the textbook. The solutions page can also be located through the bookmark on the left side (7 Percent, 7.1 Percent, Decimal, Fractions "7.3 General Applications of Percent"on pages 521–524 of the textbook. The material may also be located through the bookmark on the left side (7 Percent, 7.3 General Applications of Percent), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–4 on pages 521–524. This material contains examples of percent applications37 on pages 525–527 of the textbook. These exercises can also be located through the bookmark on the left side (7 Percent, 7.3 General Applications of Percent, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to apply concepts of percent. This includes determining the percent that you earned on a test, finding the amount of a population who fall under certain criteria, and finding the sales tax on specific items. The solutions to these problems are located in the "Answers" section on page 528 of the textbook. The solutions page can also be located through the bookmark on the left side (7 Percent, 7.3 General Applications of Percent the amount that grows or decreases by a percent.
Watching this lecture and pausing to take notes should take approximately 45 minutes.
Instructions: Please click on the link above and study the "7.4 Percent Increase or Decrease" section on pages 529–537 of the textbook. The material may also be located through the bookmark on the left side (7 Percent, 7.4 Percent Increase or Decrease), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1–6 on pages 529–537. This material contains applications of percent increase or decrease–37 on pages 538–541 of the textbook. These exercises can also be located through the bookmark on the left side (7 Percent, 7.4 Percent Increase or Decrease, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to apply the percent of increase or decrease, which includes how much an item is discounted at the store, the percent of increase in a salary, and finding the new cost of a product. The solutions to these problems are located in the "Answers" section on page 541 of the textbook. The solutions page can also be located through the bookmark on the left side (7 Percent, 7.4 Percent Increase or Decrease, "Answers").
Completing this activity should take approximately 1 hour.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the links above and complete these assignments, which test your knowledge of percent word problems. These assignments incorporate concepts of solving applications of percents. You can review the concepts associated with the questions with the Khan videos in sub-subunit 6.3.2. Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again.
The list of available graph and chart applications is endless. You may have seen applications such as trying to understand voting trends and demographics for presidential campaigns and elections. Or, a business may require graphs and charts to forecast employment growth for a specific time period. Or, you may belong to a fantasy football or baseball team, and you may need to analyze the history of points that players have against certain teams as well as other statistics. In reading a news article that provides a chart, you may want to determine what information the chart provides. Using graphs and charts is a way to convey data that is easy to understand for a specific audience. Knowing how to read and interpret these items is of utmost importance in life, because charts and graphs can be manipulated to misrepresent the data.
This unit discusses various topics when using graphs and charts in mathematics. For each type of graph in the unit, you will need to create a graph as well as interpret the results of this type of graph. You will learn to create charts and graphs (stem-and-leaf plots, line graphs, bar graphs, box-and-whisker plots, circle or pie graphs, and pictographs), read charts, and work with the measures of central tendency for a data set. (We promise it is not as scary as it sounds!)
Instructions: Please click on the links above and take notes as you watch these videos. Watch these presentations carefully two or three times until you are able to understand how to find the mean, median, mode, and range of a set of numbers.
Watching these lectures and pausing to take notes should take approximately 1 hour.
Instructions: Please click on the link above and read the "Summarizing Data" section, stopping at the "Percentiles" section, and complete the exercise associated with each topic. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice and understand terms related to the measure of central tendency exercises 1-4, 5b, 5c, 7-8. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice and understand terms related to the measure of central tendency stem-and-leaf plot.
Watching this lecture and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and read the "Stem-and-Leaf Graphs" section, stopping at the "Line Graphs" section. Then, complete Example 2. The solution to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice how to read and create a stem-and-leaf graph 1m, 3m, and 23. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to read and create a stem-and-leaf graph on reading stem and leaf plots. You can review the concepts associated with the questions with the Khan videos in subunit 7 understand how to read line graphs. Watch the second presentation carefully two or three times until you are able to understand how line graphs can be misleading.
Watching these lectures and pausing to take notes should take approximately 30 minutes.
Instructions: Please click on the link above and complete the assignment on reading line charts. You can review the concepts associated with the questions with the Khan videos in subunit 7 take notes as you watch this video. Watch this presentation carefully two or three times until you are able to understand how to find information from a bar graph.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the links above and complete these assignments, which test your knowledge of reading and creating bar graphs. These assignments incorporate concepts of reading and creating bar charts. You can review the concepts associated with the questions with the Khan videos in subunit 7.4 Watch the first presentation carefully two or three times until you are able to understand how to read a box-and-whisker plot. Watch the second presentation carefully two or three times until you are able to understand how to create a box-and-whisker plot.
Watching these lectures and pausing to take notes should take approximately 1 hour.
Instructions: Please click on the link above and read the "Box Plots" section. Then, complete Examples 1 and 2. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice how to read and create a box-and-whisker plot 5a-e, 17a, 17b, 17e, and 21. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to read and create a box-and-whisker plot assessment, which tests your knowledge on creating box-and-whisker plots. This quiz incorporates concepts of creating box-and-whisker plots, means, and quartiles. You can review the concepts associated with the questions with the Khan videos in subunit 7 take notes as you watch this video. Watch this presentation carefully two or three times until you are able to understand how to find information from a pie/circle graph.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the link above and study the "Pie Charts" section on pages 552–559 of the textbook. The material may also be located through the bookmark on the left side (7 Percent, 7.6 Pie Charts), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1 and 2 on pages 554 and 556. This material contains examples on how to interpret and represent data with a pie chart.
Reading this section and completing the exercises should take approximately 1 hour and 15 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
Instructions: Please click on the link above and complete the odd-numbered exercises for 1–29 on pages 560–565 of the textbook. These exercises can also be located through the bookmark on the left side (7 Percent, 7.6 Pie Charts, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to interpret and represent data with a pie chart. The solutions to these problems are located in the "Answers" section on page 566 of the textbook. The solutions page can also be located through the bookmark on the left side (7 Percent, 7.6 Pie Charts, "Answers").
This activity will take approximately 2 hours to complete pictograph.
Watching this lecture and pausing to take notes should take approximately 15 minutes.
Instructions: Please click on the links above and complete the assignments, which test your knowledge of interpreting pictographs. You can review the concepts associated with the questions with the Khan videos in subunit 7.7
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Math 121 -- Calculus III
Calculus III is the third semester of calculus, taken by first-year students
who have sufficient calculus background from high school or by students who have taken
Calculus II (Math 114 and either 115 or 116). It covers both differential
and integral multivariable calculus,
Informal geometric arguments take the place of formal mathematical reasoning, so
students who desire a rigorous treatment of the subject should take Math 215-216 instead.
Math 121 includes a variety of applications in all disciplines, and is appropriate
for first-year students who have not yet decided on a major.
Depending on the instructor, the mix of applications may favor either the
natural or social sciences, and students choosing this course should be sure to determine
this emphasis.
Math 121 is accessible to students who have taken only a year of calculus in high school,
yet it has a visual appeal and wealth of applications that far surpasses single variable calculus.
It is excellent preparation for many 200-level electives in mathematics,
as well as courses in chemistry and physics. It may substitute for Math 216 in the requirements for
the math major and minor and as a prerequisite for Math 317 and Math 333,
but Math 216 provides better preparation for these courses.
Prerequisites: Math 114 and either 115 or 116, or advanced placement
Who should take this course?
Students considering a major in the natural sciences, or anyone else looking for
a thorough treatment of multivariable calculus
Students considering a major or minor in mathematics who want to take multivariable calculus
before linear algebra
Students who have wish to build on their knowledge of single-variable calculus to enjoy the
visual appeal and wealth of applications of functions whose graphs are in 3-space and beyond.
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On the Study and Difficulties of Mathematics
On the Study and Difficulties of Mathematics
One of the twentieth century's most eminent mathematical writers, Augustus De Morgan enriched his expositions with insights from history and psychology. On the Study and Difficulties of Mathematics represents some of his best work, containing points usually overlooked by elementary treatises, and written in a fresh and natural tone that provides a refreshing contrast to the mechanical character of common textbooks. Presuming only a knowledge of the rules of algebra and Euclidean theorems, De Morgan begins with some introductory remarks on the nature and objects of mathematics. He discusses the concept of arithmetical notion and its elementary rules, including arithmetical reactions and decimal fractions. Moving on to algebra, he reviews the elementary principles, examines equations of the first and second degree, and surveys roots and logarithms. De Morgan's book concludes with an exploration of geometrical reasoning that encompasses the formulation and use of axioms, the role of proportion, and the application of algebra to the measurement of lines, angles, the proportion of figures, and surfaces.
Unabridged republication of the edition published by The Open Court Publishing Company, La Salle, Illinois, 1943.
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1|Traditional Portfolio
On this page you'll get a formal synopsis of me – educational history, transcripts, and so forth. But for a lot more detailed & rich information, check out the tabs above. I think you'll like what you see!
TEACHING PHILOSOPHY (a work always in progress)
Enthusiasm about math is infectious. Students learn from both discovery and direct learning. I try to constantly show my students that I care about their learning. I always try to teach to just a little higher level than my students think they can accomplish, but give them the tools and infinite encouragement to reach that level. I know, not just believe, that having clear and consistent expectations for students works.
I want to respect the privacy of my students, so I don't feel comfortable scanning in documents, but here are some projects they have done/worked on.
1. My Multivariable Calculus class does a project in the 4th quarter — on a topic of their choice. In 2009, one of my students worked on creating a harmonograph which draws damped Lissajous Curves. He then explained the theory behind the harmonograph, and wy the parametric equations being drawn took the form that they did. A video of the Harmonograph is below.
2. For our Algebra II classes in 2007/8, the other Algebra II teacher and I created a "video project" which was used to promote student communication in mathematics. Each student signed up for a topic and created a video "teaching" the topic (with an example) to the rest of the students in Algebra II. They were put up on a blog called Logarithms, Rational Functions, and Trigonometry! Oh My! Feel free to click on the name to visit it, or to visit the post mortem (analysis) on my blog here.
EXAMPLES OF MY WORK
I teach most of my classes using SmartBoard, sometimes alongside worksheets I create.
1.You can read my paean to Smartboard here. But you probably just want to see samples.
2. In 2008/9, when studying quadratic regressions, my Algebra II class collected some data from pendulums of various lengths and analyzed them. The original idea for the "lab" is here, and a blow-by-blow for the lab, our thought processes, and our conclusion, is here.
3. In 2008/9, in Algebra II, I made a series of packets to teach linear and quadratic inequalities. The topic list is here. The packets are:
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Linear Algebra : Introduction - 2nd edition
Summary: In this appealing and well-written text, Richard Bronson gives readers a substructure for a firm understanding of the abstract concepts of linear algebra and its applications. The author starts with the concrete and computational, and leads the reader to a choice of major applications (Markov chains, least-squares approximation, and solution of differential equations using Jordan normal form).
The first three chapters address the basics: matrices, vector s...show morepaces, and linear transformations. The next three cover eigenvalues, Euclidean inner products, and Jordan canonical forms, offering possibilities that can be tailored to the instructor's taste and to the length of the course. Bronson's approach to computation is modern and algorithmic, and his theory is clean and straightforward. Throughout, the views of the theory presented are broad and balanced. Key material is highlighted in the text and summarized at the end of each chapter. The book also includes ample exercises with answers and hints. With its inclusion of all the needed features, this text will be a pleasure for professionals, teachers, and36.00 +$3.99 s/h
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PAPERBACK New 0120887843 Excellent condition. Never been used. **FREE** Delivery tracking with every book purchased.
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Created by Jill Stevens of Illuminations: Resources for Teaching Mathematics, this activity allows students to look for functions within a given set of data. After analyzing the data, the student should be able to...
This online, interactive lesson on distributions provides examples, exercises, and applets which explore the basic types of probability distributions and the ways distributions can be defined using density functions,...
Exercises posted on this web site offer an opportunity for students to evaluate how much they have retained in various subjects of Algebra. Topics covered include geometry, functions, vectors, and statistics. There are...
Statistical Associates Publishing is a creation of Professor Dave Garson and hosts a number of free statistics e-books, and some low-cost Kindle versions as well. Use of the site is password-protected, so visit the...
This unit from Illuminations focuses on collecting data and using technology to find functions to describe the data collected. Students will learn to use a calculator to find the curve of best fit for a set of data and...
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Intermediate Algebra Connecting Concepts Through Applications
Description: INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problemMore...
INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master concepts, problem solving, and communication skills. It modifies the rule of four, integrating algebraic techniques, graphing, the use of data in tables, and writing sentences to communicate solutions to application problems. The authors have developed several key ideas to make concepts real and vivid for students. First, the authors integrate applications, drawing on real-world data to show students why they need to know and how to apply math. The applications help students develop the skills needed to explain the meaning of answers in the context of the application. Second, they emphasize strong algebra skills. These skills support the applications and enhance student comprehension. Third, the authors use an eyeball best-fit approach to modeling. Doing models by hand helps students focus on the characteristics of each function type. Fourth, the text underscores the importance of graphs and graphing. Students learn graphing by hand, while the graphing calculator is used to display real-life data problems. In short, INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS takes an application-driven approach to algebra, using appropriate calculator technology as students master algebraic concepts and
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CKSD calculating new math requirements
The Klahowya Secondary School ninth-grader's feet can't quite reach the ground when he sits in one of the heavy, cushy chairs in the main office. When the red-haired freshman was in sixth grade, upon teacher recommendation he sped up his math studies and by seventh grade, he was in algebra 1.
"Normally I'd be in geometry at this age," said Kevin Hassett who is enrolled in the first all-freshman algebra 2 class at KSS.
Hassett says he's always been a fan of functions and sequences and sitting in Ellen Kraft's algebra 2 classroom has proven to be a complementary occupation to his advanced physical science class.
One reason he takes math is because he looks ahead to where the numbers could figure into his career.
"I think in my future jobs which might include engineer or something in the realm of science (math would be useful) and again I take it because I like it," Hassett said.
An appreciation of math like Hassett's adds up to perfection in Dave Thielk's
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This best-selling guide from authors Elaine Weinmann and Peter Lourekas has been the go-to tutorial and reference book for photography/design professionals and the textbook of choice in college classrooms for decades. This edition includes their trademark features of clear, concise, step-by-step instructions; hundreds of full-color images; screen captures of program features; and supplemental tips and sidebars in every chapter.
Practice makes perfect-and helps deepen your understanding of algebra 1,001 Algebra I Practice Problems For Dummies, with free access to online practice problems, takes you beyond the instruction and guidance offered in Algebra I For Dummies, giving you 1,001 opportunities to practice solving problems from the major topics in algebra. You start with some basic operations, move on to algebraic properties, polynomials, and quadratic equations, and finish up with graphing.
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Math books
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Essential Math With Application - 8th edition
Summary: The latest book from Cengage Learning on Essential Mathematics As in previous editions, the focus in ESSENTIAL MATHEMATICS with APPLICATIONS remains on the Aufmann Interactive Method (AIM). Users are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. The role of ''active participant'' is crucial to success. Presenting students with worked examples, and then providing ...show morethem with the opportunity to immediately work similar problems, helps them build their confidence and eventually master the
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Series
Lessons: 12
Total Time: 0h 59m
Use: Watch Online & Download
Access Period: Unlimited
Created At: 06/29/2011
Last Updated At: 06/29/2011
This 11-lesson video series will walk you through several applications of quadratic functions and equations. We'll look at some word problems, some quadratic geometry problems, some problems that ask us to find minimum and maximum points using quadratic functions, etc. To learn about these concepts, which are critical to Algebra, we'll cover the topic through a series of video lessons, each of which will cover pertinent ideas and related problems. The video content in this series will include a lesson on each of the following:
* Word Problems Involving Quadratics, Ex 1
* Word Problems Involving Quadratics, Ex 2
* Word Problems Involving Quadratics, Ex 3
* Maximum Values of Quadratic Functions
* Find Max & Min of a Quadratic Function
* Solving a Quadratic Geometry Problem, Ex 1
* Solving a Quadratic Geometry Problem, Ex 2
* Solving a Quadratic Geometry Problem, Ex 3
* The Pythagorean Theorem, Ex 1
* The Pythagorean Theorem, Ex 2
* The Pythagorean Theorem, Ex 3
These videos are available to be viewed online for free on PatrickJMT
About this Author
Many of the videos for sale are also available on my website: or you can also do a search and check out my popular 'math channel' on YouTube (PatrickJMT). You can watch all of them there for FREE!
Masters degree in Mathematics; former math instructor at a top 20 university!
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Beginning Algebra
9780495118077
ISBN:
0495118079
Edition: 8 Pub Date: 2007 Publisher: Thomson Learning
Summary: Easy to understand, filled with relevant applications, and focused on helping students develop problem-solving skills, BEGINNING ALGEBRA is unparalleled in its ability to engage students in mathematics and prepare them for higher-level courses. Gustafson and Frisk's accessible style combines with drill problems, detailed examples, and careful explanations to help students overcome any mathematics anxiety. Their prove...n five-step problem-solving strategy helps break each problem down into manageable segments: analyze the problem, form an equation, solve the equation, state the conclusion, and check the result. Examples and problems use real-life data to make the text more relevant to students and to show how mathematics is used in a wide variety of vocations. Plus, the text features plentiful real-world application problems that help build the strong mathematical foundation necessary for students to feel confident in applying their newly acquired skills in further mathematics courses, at home or on the job.
Gustafson, R. David is the author of Beginning Algebra, published 2007 under ISBN 9780495118077 and 0495118079. Eighty Beginning Algebra textbooks are available for sale on ValoreBooks.com, seventy two used from the cheapest price of $0.53, or buy new starting at $111.58.[read more heavy shelf & corner wear, but still a good reading copy. Does not include Online Resource We are a tested and proven company with over 900,000 satisfied customers si [more]
Has heavy shelf & corner wear, but still a good reading copy. Does not include Online Resource We are a tested and proven company with over 900,000 satisfied customers since 1997. Choose expedited shipping (if available) for much faster delivery. Delivery confirmation on all US orders Instructor's Edition. Like new hardcover 8th edition, NO ACCESS CARD. Identical to Student Edition only contains notes and solutions to all problems.Shipping from Ca [more]
ALTERNATE EDITION: Annotated Instructor's Edition. Like new hardcover 8th edition, NO ACCESS CARD. Identical to Student Edition only contains notes and solutions to all problems.Shipping from California.[less]
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Written in a readable, yet mathematically mature manner appropriate for college algebra level students, Coburn's Precalculus uses narrative, extensive examples, and a range of exercises to connect seemingly disparate mathematical topics into a cohesive whole.
With the same design and feature sets as the market leading Precalculus, 8/e, this addition to the Larson Precalculus series provides both students and instructors with sound, consistently structured explanations of the mathematical concepts. Designed for a two-term course, this text contains the features that have made Precalculus a complete solution for both students and instructors: interesting applications, cutting-edge design, and innovative technology combined with an abundance of carefully written exercises. In addition to a brief algebra review and the core precalculus topics, PRECALCULUS WITH LIMITS covers analytic geometry in three dimensions and introduces concepts covered in calculus.
Mathematics Describing the Real World Precalculus and Trigonometry-Bruce H. Edwards AVI, XviD, 640x480, 29.97 fps | English, MP3@128 kbps , 2 Ch | ~36x30 mins | 10.82 GB The Teaching Company | 2011 | Course no. 1005 Trad... Filesonic, Fileserve, Uploading, Wupload, Uploadstation Links Engoy all members !!!...
TradClear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text like you. The book also provides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics.
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CLE 3102.1.3 Develop inductive and deductive reasoning to independently make and evaluate mathematical arguments and construct appropriate proofs; include various types of reasoning, logic, and intuition.
CLE 3102.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies.
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Emeryville Math these tests and the new common core standards. Algebra 1 consists of symbolic reasoning and calculations with symbols. Through
the study of algebra, a student develops an understanding of the symbolic language
of mathematics and the sciencesIn the conventional education system today memorization is stressed over comprehension, but for professionals to be successful in the increasingly competitive work world today, there needs to be more than just rote drilling and memorization--there needs to be a true appreciation and comprehension...
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Well you could cartainly try. This kind of thing was tried back in the 60s, it didn't get very far. There is no point in designing a curriculum that only targets a small minority of people.
The basic fact is most people in the world are dreadful at at mathematics. This is something this forums seems to overlook, people who acutally enjoy physics and mathematics are very much in the minority in this world. It's largely pointless in teaching children something that will serve them no purpose in later life. Statistics would serve the general denizens of this world far better than calculus ever could.
I think people tend to be dreadful at mathematics because it is not taught properly. Calculus isn't just useful for analyzing physics and chemistry and assorted other topics, it has explanatory value in and of itself. Again speaking from personal experience, algebra became so much clearer once I knew why those techniques were taught. For instance, I couldn't give a darn less about multiplying by conjugates until I started studying limits. I didn't appreciate the beauty of e until I heard the definition of the natural logarithm. I certainly didn't much understand the purpose of being able to discern the equation of a line from a point and a slope. To me, learning calculus is like a myopic person putting on glasses. Sure, you can vaguely understand some concepts, but it's so much clearer in light of calculus.
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Foundations of Geometric Algebra Computing
This book lays the foundation for the widespread use of geometric algebra as a new, up-and-coming field of geometrically intuitive and performant computing technology with a wide range of potential engineering applications in academia and industry.
The The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics.This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications.The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.
Table of Contents
Table of Contents
Chap. 1 Introduction.
Chap. 2 Mathematical Introduction.
Chap. 3 The Conformal Geometric Algebra.
Chap. 4 Maple and the Identification of Quaternions and Other Algebras.
Chap. 5 Fitting of Planes or Spheres into Point Sets.
Chap. 6 Geometric Algebra Tutorial Using CLUCalc.
Chap. 7 Inverse Kinematics of a Simple Robot.
Chap. 8 Robot Grasping an Object.
Chap. 9 Efficient Computer Animation Application in CGA.
Chap. 10 Using Gaalop for Performant Geometric Algebra Computing.
Chap. 11 Collision Detection Using the Gaalop Precompiler.
Chap. 12 Gaalop Precompiler for GPGPUs.
Chap. 13 Molecular Dynamics Using Gaalop GPC for OpenCL.
Chap. 14 Geometric Algebra Computers
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payment types
certifications, Sixth Edition was written to help readers effectively make the transition from arithmetic to algebra. The new edition offers new resources like
Find more:Find more:
Highlights:
Created for the independent, homeschooling student, Teaching Textbooks has helped thousands of high schoolers gain a firm foundation in upper-level math without constant parental or teacher involvement. Extraordinarily clear illustrations, examples, and graphs have a non-threatening, hand-drawn look, and engaging real life questions make learning pre-algebra practical and applicable. Textbook examples are clear while the audiovisual support includes lecture, practice and solution CDs for every chapter,
Highlights: & Introductory Algebra, Third Edition was written to help readers effectively make the transition from arithmetic to algebra. The new editionSolid preparation for algebra and geometry. Integers and algebraic concepts are introduced beginning in Chapter 1 to develop algebraic thinking skills. Throughout the text, connections are made to arithmetic skills. Geometry concepts are integrated when appropriate to foster connections. With an emphasis on the mastery of basic skills, the text provides numerous opportunities to assess progress in basic skills along with abundant remediation and intervention activities.</p>
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" 9.63 teaches principles of experimental methods in human perception and cognition, including design and statistical analysis. The course combines lectures and hands-on experimental exercises and requires an independent experimental project. Some experience in programming is desirable. To foster improved writing and presentation skills in conducting and critiquing research in cognitive science, students are required to provide reports and give oral presentations of three team experiments. A fourth individually conducted experiment includes a proposal with revision, and concluding written and oral reports."
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This is an introductory course to MATLAB, the high-performance interactive software. Topics ...
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This is an introductory course to MATLAB, the high-performance interactive software. Topics include MATLAB Basics, Plotting, Scripts & Functions and Programming. Additional resources are also provided.
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In this section, we will go over the Matlab code we used to simulate our project, the various algorithms we tried, how we simulated "real-time", and how the matlab simulation dealt with real signals.
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Introduces the fundamentals of machine tool and computer tool use. Students work ...
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Introduces the fundamentals of machine tool and computer tool use. Students work with a variety of machine tools including the bandsaw, milling machine, and lathe. Instruction given on the use of the Athena network and Athena-based software packages including MATLABĺ¨, MAPLEĺ¨, XESSĺ¨, and CAD. Emphasis on problem solving, not programming or algorithmic development. Assignments are project-oriented relating to mechanical engineering topics. It is recommended that students take this subject in the first IAP after declaring the major in Mechanical Engineering. From the course home page: This course was co-created by Prof. Douglas Hart and Dr. Kevin Otto.
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Numerical Computing with MATLAB is a textbook for an introductory course in ...
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Numerical Computing with MATLAB is a textbook for an introductory course in numerical methods, MATLAB, and technical computing. It emphasizes the informed use of mathematical software. Topics include matrix computation, interpolation and zero finding, differential equations, random numbers, and Fourier analysis.
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Free Printable Algebra 1 Worksheets - Also Available Online
There are a number of free algebra 1 worksheets for you to download, print, or solve online. The worksheets cover evaluating equations, exponents addition, inequalities, multiplication of exponents, and solving algebra equations in a minimal amount of steps.
Begin by selecting the free algebra worksheet you would like to have. This will take you to the web page of the algebra 1 worksheet. You then have several options. You can print the worksheet, download the corresponding PDF file, or complete the free algebra worksheet online. The online feature works as long as you are using a modern web browser, your iPad or other tablet device. Now you are all ready to start solving algebra equations.
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Jump right in! Anybody and everybody is welcome to help others! It is not required that you be a professor, an accredited educator, or a professional tutor. All you need is a familiarity with a topic and the ability to communicate your knowledge.
Encourage learning. The goal of these forums is to help people learn and, as they say, "One best learns by doing." While it is often easiest to simply post "the answer" to an exercise, a student learns nothing from this that wasn't already in the back of the book. Instead, strive to provide hints, helps, leading questions, and other avenues for the learner to investigate.
Support growth. If you're helping, then you already "know how to do it". The goal of these forums is that the learners grow in understanding as you have. Providing the complete hand-in solution to a take-home test might be "fun" for you, but the student (experience shows) learns very little which is positive.
Model good habits. Please speak clearly, display consideration and good manners, and model useful mathematical habits of writing and thought. Explaining similar examples and posting links to on-topic web pages is great; doing a student's homework or posting ads is not.
Be professional. Keep in mind that readers cannot hear your cheerful tone or see your friendly wink. While a friendly tone is certainly preferred, it is often best to exercise caution regarding sarcasm, off-topic humor, and the like. If you're tired, go to bed and come back, refreshed, the next day.
Please keep in mind that the purpose of these forums is to help students learn and grow, so let's try to keep current threads on-topic. And, as excellent a suggestion as you might have for a particular question, resurrecting an old (and especially a resolved) thread accomplishes little more than pushing active questions out of view, so let's stick to responding to recent and unresolved threads.
I am reviewing my calculus and I stumbled over this forum just today by googling for some help on rate-of-change of an angle type word problems. We worked together on another virtual venue some time back. I am really glad to see that you have created this forum and I will try to help when I can.
Cinnamon29 wrote:If you have specific questions then post them in the appropriate sub-forum and someone will help you.
These forums are a fantastic idea, no website I have ever visited has got a forum. It really helps and if you are stuck on something you can just post your problem and people will answer there best ideas. Amazing.
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Math utilities
Vectors and Matrices The vector and matrix classes provide commonly used mathematical objects and algorithms which include:
- Basic calculations involving vectors and matrices
- Computation of eigenvalues and eigenvectors of a square matrix
- Solvers for a set of linear algebraic equations
- Algorithms to find the roots of a set of nonlinear equations
- Algorithms to find the minimum function of one or more independent variables
These classes also provide a data structure in order to represent any expression, relation, or function used in mathematics, including the assignment of variables.
gp also provides means for positioning geometry in space or on a plane using an axis or coordinate system, and defines the following geometric transformations:
- Translations
- Rotations
- Symmetries
- Scaling transformations
- Composed transformations
Common Math Algorithms
The common math algorithms provided in Open CASCADE Technology include:
- Algorithms to solve a set of linear algebraic equations
- Algorithms to find the minimum of a function of one or more independent variables
- Algorithms to find roots of one or of a set of non-linear equations
- An algorithm to find the eigenvalues and eigenvectors of a square matrix
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A unique approach illustrating discrete distribution theory through combinatorial methods This book provides a unique approach by presenting combinatorial methods in tandem with discrete distribution theory. This method, particular to discreteness, allows readers to gain a deeper understanding of theory by using applications to solve problems. The... more...
This book provides explanatory text, illustrative mathematics and algorithms, demonstrations of the iterative process, pseudocode, and well-developed examples for (familiar as well as novel) applications of the branch-and-bound paradigm to relevant problems in combinatorial data analysis. more...
A collection of the various old and new results, centered around the following simple observation of J L Walsh. This book is particularly useful for researchers in approximation and interpolation theory. more...
Mathematics for Dyslexics: Including Dyscalculia, 3rd Edition discusses the factors that contribute to the potential difficulties many dyslexic learners may have with mathematics, and suggests ways of addressing these difficulties. The first chapters consider the theoretical background. The later chapters look at practical methods, which may help dyslexic... more...
Shows how well-meant teaching strategies and approaches can in practice exacerbate underachievement in maths by making inappropriate demands on learners. As well as criticizing some of the teaching and grouping practices that are considered normal in many schools, this book also offers an alternative view of attainment and capability. more...
Veteran educators share proven solutions to guide a new secondary math teacher through the challenging first few months and provide the more experienced teacher with interesting alternatives to familiar methods
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Calcula = THE CALCULATOR ... but not limited to the calculator. Calcula is not a scientific calculator. Calcula is a tool 'all-in-one': instead of having 1, 2, 5, 10, 20 applications that serve as 'technical means' we have only one: Calcula, indeed!So, what is and what makes Calcula?. calculator with the 4 operations, percentage, square root, exponentiation of x, a fraction of 1, form, and factor accumulation and subtraction in memory ... what they do all the calculators, some (not quite all, actually!). storing the list of all the transactions like a roll of paper with its zoom. button to cancel last input CI, C key to cancel the entire operation and key 'tearing paper' to delete all memorized transactions. selection of the number of decimal places, from 0 to 5, with which to develop. currency conversion online, in real time and then, leaning on a free service of common good (the result can be integrated in the operation in progress). conversion between many units of measurement: length, weight, volume, area, etc.. (The result can be integrated in the operation in progress). conversion between different number systems: decimal, octal, hexadecimal and binary (the result, of course decimal) can be integrated in the operation in progress). calculating perimeter, area and volume of many geometric shapes with a list of requests for images and input context to the figure (within the perimeter of the circle or to calculate the volume of the cylinder, or more or less according to your traps, etc..)(The result can be integrated in the operation in progress). expression processing up to 26 variables and many functions available, such as cos, sin, tan, etc.. (The result can be integrated in the operation in progress). development of algebraic proportions of the type: b = x: c-fit of the 3 known values and the processing of the result in 4 combinations (the result can be integrated in the operation in progress). generation of random numbers indicating the amount of numbers to be generated and the minimum and maximum limits (ability to select whether the numbers generated should all be different or with repetitions). elaborations of summations, differences between dates with even numbers add or subtract days. elaboration of summations, differences between zones with even add or subtract a preset time. stopwatch with lap times list the possibility of. flashlight (beam) with a selection of different colors. in cm and inch ruler, and color-changing background and calibration lines for even better viewing of the backlit. compass needle or rotary dial with digital indication of the degree. level graphics and digital indication of the degree of vertical tilt and horizontal. selection if it beeps when you press any buttons or voice with repetition of numbers typed and conducting operations in. ability to change the background color. appropriate option for the configuration settings. Detailed help on all aspects. Calcula the program is released with 2 screens, others are making and will be issued free of charge even after the purchase) to have more or fewer buttons then more or less the same size buttons. In version 1.1.00 there are 2 screens: the no. 0 with all the buttons available, some with 2 or 3 functions enabled via special button shift, the no. 1 instead of the calculator and all transactions with a button that serves as a menu to call up all the other functions.. all the screens are operated by the minimum resolution is 320x480 portrait or landscape (480x320). ON / OFF switch!The program is released in Italian, English
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Microsoft student with Encarta premium 2007: Microsoft Corp.
Encarta provides comprehensive resources to help students complete homework assignments in math, science, language artslanguage arts pl.n. The subjects, including reading, spelling, and composition, aimed at developing reading and writing skills, usually taught in elementary and secondary school. , foreign language and social studies. It includes multimedia encyclopedia brand, graphing calculatorGraphing Calculator may refer to:
Graphing calculators, calculators that are able to display and/or analyze mathematical function graphs.
NuCalc, a computer software program able to perform many graphing calculator functions.
software and step-by-step assistance on equations ranging from pre-algebra through calculuscalculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value. and sciences. Another new product. Learning Essentials for Microsoft Office Microsoft's primary desktop applications for Windows and Mac. Depending on the package, it includes some combination of Word, Excel, PowerPoint, Access and Outlook along with various Internet and other utilities. 1.5. a free download with a volume license, customizes Microsoft Office applications. While it offers more than 180 curriculum-based templates and 65 tutorials, teachers can convert documents into standards-based learning resources to share, search and use with any learning management system that complies with Shareable Content Object Reference Model 1.2 or 2004 standards.
COPYRIGHT 2006 Professional Media Group LLC
No portion of this article can be reproduced without the express written permission from the copyright holder.
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0321569288
9780321569288
Videos on DVD with Optional Subtitles for A Problem Solving Approach to Mathematics for Elementary School Teachers:The Video Lectures on DVD provide a lecture for each section of the textbook. Video lectures cover important definitions, procedures and concepts from the section by working through examples and exercises from the textbook. Videos have optional subtitles.
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Rent Videos on DVD with Optional Subtitles for A Problem Solving Approach to Mathematics for Elementary School Teachers 10th edition today, or search our site for Rick textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
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