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Highlight the language of domain and range, and the ideas of continuity and discontinuity, with this tool that links symbolic and graphic representations of each interval of a piecewise linear functio... More: lessons, discussions, ratings, reviews,...
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On this online calculator calculate mathematical expressions and complex numbers. You can do matrix algebra and solve linear systems of equations and graph all 2D graph types. You can also calculate z... More: lessons, discussions, ratings, reviews,...
Plomplex is a complex function plotter using domain coloring. You can compose a function with a complex variable z, and generate a domain coloring plot of it. You can choose the plot range as well as ...
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Category Archives: 16. Indices and Standard Form
Overview This covers; Laws of Indices, Fractional and Negative Powers, Standard Form (building on knowledge gained in Complex Calculations and Accuracy) and Surds. Some of this is quite complicated (some Grade A and A* questions). I think the trick is … Continue reading →
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This Page
Undergraduate Students
The Mathematical Association of America (MAA) is the largest
mathematical society in the world that focuses on mathematics for
students, faculty, professional mathematicians, and all who are
interested in the mathematical sciences; that is, mathematics at the
undergraduate level. Our members include university, college, and high
school teachers; high school, undergraduate and graduate students; and
others in academia, government, business, and industry. Our core
interests are Education, Research, Professional
Development, Public Policy, and Public
Appreciation. The student web pages cover topics in academics,
careers, research/summer opportunities, meetings for students, and
more. If you are not yet a member, we urge you to consider joining and
ask you to visit our membership
page. As a member, you can help the MAA fulfill our goals to
benefit you, the student.
Enrollment in your regional MAA section, where you can catch up
with professors and other mathematics students, hear some good talks
about mathematics, give a talk yourself or participate in a special
session for students.
The William Lowell Putnam Mathematical Competition is an annual contest
for college students established in 1938 in memory of its namesake.
Cash prizes for the top five teams in recent years ranged from $25,000
to $5,000. Recent cash prizes for the top five individuals have been
$2,500 each. The exam, administered by the MAA, is held the first
Saturday in December (Note: registration
needs to be in the Director's hands by mid-October). More information (including how to
register for the exam) can be found at the official site.
Mathematicians find careers in many diverse fields. The most popular
career choice for Mathematicians is teaching, which many find to be
very rewarding. At the same time, one of the top ten jobs every year is
actuary, which applies statistics to determine the chance of risk and
its financial consequences. Mathematicians work in operations
research, computer science, cryptography, biotechnology, and more.
Visit the MAA Careers page
for more information.
Ready to look for a job? MAA
Math Classifieds is available to help you find a career in the
diverse field of mathematics. We invite you to explore this site to
begin your job search.
Over the last few decades, there has been an explosion in
undergraduate student research. Much of this is being done through
faculty-directed research within a department while others are done in
summer research programs (see below). Student research is the ultimate
in engaged learning. Sometimes the result of a capstone course,
sometimes because a school requires a Senior Thesis, and sometimes
resulting from an enthusiastic student approaching a professor and
asking for help, these research experiences are proven to enhance the
student's time in school. In a survey by David Lopatto of Grinnell
College "students rated benefits such as 'learning a topic in depth,'
'developing a continuing relationship with a faculty member,'
'understanding the research process in your field,' and 'readiness for
more demanding research' very highly." Undergraduate research can lead
to speaking at local and/or national meetings or publication is
research journals.
Budapest
Semester in Math - A 15-week mathematics study abroad program in
Budapest, Hungary. Students take mathematics classes taught in English.
Junior Year for Women at
Smith College - For women in the junior year to attend Smith
College to get intensive training in mathematics while building the
skills and confidence needed to succeed in graduate school mathematics.
Math in Moscow - A
15-week mathematics study abroud program in Moscow, Russia. Students
take math classes taught in English.
Research
Experience for Undergraduates- REU Programs are summer
programs sponsored by the National Science Foundation (NSF). REUs
usually consist of two parts: intensive study of topics through lecture
and interaction, and student research on a question/questions. Travel
costs are paid for as well as room and board. A stipend is given to
participants. These are all available on a competitive basis.
National Research
Experience for Undergraduates Program - The NREUP is
administered by the MAA through its SUMMA (Strengthening
Underrepresented Minority Mathematics Achievement) Program and made
possible via a grant from the Moody's Foundation, the NSF, and the
National Security Agency. These research experiences are similar in
nature to REUs, but are designed to reach out to minority students at
the midpoint of their undergraduate programs.
Summer Program for
Women in Mathematics - Hosted by George Washington University, this
intensive five-week program is for women who have finished their junior
year of college and may be considering graduate work in the
mathematical sciences.
Meetings are an important part of the educational process. These are
where you can meet other people with your same interest in mathematics.
It's where you can listen to talks on your favorite subject, see a
panel discussion on careers, and relax watching Math Jeopardy.
Sections Meetings
– the MAA divides the U.S. and Canada into 29 sections, each of
which have at least one meeting per year.
There could be one near you soon.
Joint
Mathematics Meetings - The JMM is a combined meeting of the
MAA along with the American Mathematics Society (AMS) and others, held
every January. This is a large gathering, drawing more than 5000
participants per year (including hundreds of students like you). It
sounds intimidating, but there are many things for undergraduates to
do. One highlight is the
Undergraduate Student Poster Session, where undergraduates present
their research. An article covering the 2009 Poster Session has been
published in MAA Focus.
The MAA also publishes a brochure for students on how to get the most out of the JMM
and MathFest.
MathFest - The large
national summer meeting of the MAA. Its concentration is on students,
some talking about their research they have been involved in throughout
the year and the summer, others enjoying Math Jeopardy, and still more
taking advantage of activities such as a session on math and origami.
While not as big as the JMM, it is a great
end-of-summer/beginning-of-school-year event. Additionally, there are
funds available to help with the travel cost of those who will be
speaking. Click here for advice about applying for travel funding.
The natural questions that come up, "Is graduate school what I
want?" and "How can an M.S. or Ph.D. help me?" can start to be answered
by looking that the pages here
and here
Master's Degree
Perhaps you've found a favorite subject that you want to investigate
more deeply. Maybe you wish to add to your undergraduate degree and
make yourself more marketable. Whatever the reason, there are plenty of
options for Master's degree programs. At most of them, you'll qualify
for a Teaching Assistanship or Research Assistantship which will help
pay for the program. It's an investment which can have a big return.
According to the Census Bureau, a Master's degree is worth almost
$500,000 more over a working lifetime than a Bachelor's alone.
Professional Master's Degree
What is a professional Master's? It is a degree which is not meant
to feed into a PhD program, but is capable of standing on its own.
Terminal Master's degrees in mathematics can be found with
concentration in fields such as Biology, Finance, and Operations
Research. For a good list of subjects, and schools with professional
Master's in that subject, go to the AMS page.
PhD
A PhD in mathematics usually brings to mind a career teaching at a
university. While this is true, there are a lot more opportunities
available. Operations research analyst, statistician, biomathematician,
and more are available. AMS has put together a good presentation
on finding an industry job. For advice on both non-academic and academic jobs, check out
the AMS's Advice for New PhDs.
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College Trigonometry - 6th edition
Accessible to students and flexible for instructors, College Trigonometry, Sixth Edition, uses the dynamic link between concepts and applications to bring mathematics to life. By incorporating interactive learning techniques, the Aufmann team helps students to better understand concepts, work independently, and obtain greater mathematical flu...show moreency. The text also includes technology features to accommodate courses that allow the option of using graphing calculators. Additional program components that support student success include Eduspace tutorial practice, online homework, SMARTHINKING Live Online Tutoring, and Instructional DVDs.
The authors' proven Aufmann Interactive Method allows students to try a skill as it is presented in example form. This interaction between the examples and Try Exercises serves as a checkpoint to students as they read the textbook, do their homework, or study a section. In the Sixth Edition, Review Notes are featured more prominently throughout the text to help students recognize the key prerequisite skills needed to understand new concepts.
Updated! End-of-chapter exercises--Assessing Concepts--have been revised to include more question types including fill-in-the-blank, multiple choice, and matching.
Revised!Prepare for This Section exercises, formerly Prepare for the Next Section, have been moved from the end of each chapter to the beginning of each chapter and afford students the opportunity to test their understanding of prerequisite skills about to be covered.
New!Calculus Connection icons have been added to indicate topics that will be revisited in subsequent courses, laying the groundwork for further study.
Applications require students to use problem-solving strategies and new skills to solve practical problems. Covering topics from many disciplines, including agriculture, business, chemistry, education, and sociology, these problems demonstrate to students the practicality and value of algebra.
Noted by a pie chart icon, Real Data examples and exercises require students to analyze and construct mathematical models from actual situations.
Appearing throughout the text, Integrating Technology notes offer relevant information about using graphing calculators as an alternative way to solve a problem. Step-by-step instructions allow students to use technology with confidence.
Exploring Concepts with Technology, an optional end-of-chapter feature, uses technology (graphing calculators, CAS, etc.) to explore ideas covered in the chapter. These investigations can be used in a variety of ways, such as group projects or extra-credit assignments. Together with Integrating Technology tips, this feature makes the text appropriate for courses that allow the use of graphing calculators.
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Page Quick Links
About Grades 9–12 Mathematics
Denver Public Schools' vision for high-quality mathematics instruction embraces the technological advances of the 21st century: global commerce and the ubiquitous use of computers. The Mathematics Department wants to ensure that students are equipped to qualify for employment in a global economy by preparing them to "absorb new ideas, perceive patterns, and solve unconventional problems"
To meet this challenge, DPS math educators designed a program that moves students beyond collecting and recollecting facts and performing memorized procedures. The program reflects dramatic changes that have occurred in mathematics and capitalizes on our understanding of how students learn. Ultimately, the program guides students through a discipline that has practical and useful outcomes: employment in a world where a solid understanding of mathematics can help pave the way to employment success.
Goals for Denver Students of Mathematics
Solve problems when no routine path to the answer is apparent.
Use mathematical signs, symbols, and terms to communicate ideas and to clarify, refine, and consolidate mathematical thought processes.
Reason mathematically by moving through the process of gathering data, making conjectures, assembling evidence, and supporting or refuting conjectures.
Make mathematical connections among seemingly unrelated ideas or concepts through holistic portrayal of the role mathematics plays in our world.
View mathematics as a common human activity that helps make sense of any problem situation.
Appreciate and enjoy an exploratory view of the world through mathematics.
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These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's Precalculus: Enhanced with Graphing Utilities gives students a model for success in mathematics.
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Karma reasons for concrete message
Message
Honestly, there's a better solution to this issue. The question should be tackled from a different angle: how is algebra (and all the maths) taught???
My wife is a math teacher. She loves the actual teaching part--WHEN SHE'S ALLOWED TO DO IT THE WAY IT SHOULD BE DONE. But with state- and federally-mandated testing of a huge variety of topics, teachers are forced to cram as much information as they can down students' throats. They *must* "teach to the test" in order to keep their schools alive.
Teachers don't get the opportunity to give the students THEIR opportunity to understand the concepts behind the algorithms. They are so time-crunched with the amount of material they need to cover that there isn't time to insure kids understand the "why," so they are obligated to get them to rote-memorize the "what and how" so they can pass their test.
Allow teachers to use the methods they're trained at, are expert in. Allow teachers to put responsibility on the students and parents (another topic entirely, but a very salient one!) for the students' learning. Allow teachers to use research-based teaching methods, rather than just rote memorization "because that's the way it's always been done." Let teachers effing TEACH. THEN this whole question will likely go away, because kids will be able to LEARN algebra--to UNDERSTAND it.
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Algebra 2 is a lot harder than it used to be. It's also more important than it used to be because algebra 2 concepts are included on the new SAT. Sadly, most of today's students have very limited vocabularies.
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Course Description
MA 1000
Foundations of College Mathematics
This course prepares students to take MA 1010 or MA1025. Topics include: computation with integers and rational numbers, using correct order of operations, ratios, and proportions. The student also learns percent concepts and solving equations involving percentages. Other topics covered include exponents, simplifying and solving equations, and inequalities with one variable. Linear equation problem solving strategies are emphasized in the solution of application problems Problem solving is integrated throughout and appropriate use of calculators is expected.
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Nature
and Role of
Algebra in
the K-14
Curriculum
Proceedings ofa
NalionaISymposium
May 27 and 2S, 1997
Sponsored by
National Council of Teachers of Mathematics and
Mathematical Sciences Education Board
Center for Science, Mathematics, and Engineering Education
National Research Council
National Academy Press
Washington, D.C. 1998
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NATIONAL ACADEMY PRESS · 2101 Constitution Avenue, NW · Washington, DC 20418
NOTICE: The project that is the subject of this report was approved on February 21, 1997, by the Executive Committee of the
Governing Board of the National Research Council (NRC), whose members are drawn from the councils of the National
Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The project was approved on April
22, 1996, by the Board of Directors of the National Council of Teachers of Mathematics (NCTM).
This report has been reviewed by a group other than the authors according to procedures approved by a Report Review
Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute
of Medicine.
Additional copies of this report are available from the National Academy Press,2101 Constitution Avenue, NW, Lock Box 285,
Washington, DC 20055. (800) 624-6242 or (202) 334-3313 (in the Washington metropolitan area).
International Standard Book Number 0-309-06147-4
This report is available online at and
Printed in the United States of America.
at
Except as otherwise noted, copyright 1998 by the National Academy of Sciences. All rights reserved.
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NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS
The National Council of Teachers of Mathematics (NCTM), founded in 1920, is a nonprofit professional
association dedicated to the improvement of mathematics education for all students in the United States and
Canada. It offers vision, leadership, and avenues of communication for those interested in the teaching and learning
of mathematics at the elementary-school, middle-school, high-school, college, and university levels. With more
that 110,000 members, NCTM is the largest mathematics education organization in the world. Each year, the
NCTM conducts a large national conference and seven to nine regional conferences, where teachers of mathematics
and others interested in mathematics education can attend lectures, panel discussions, and workshops and can see
exhibits of the latest mathematics education materials and innovations. Many NCTM members are also members of
one or more of the 260-plus local and special-interest groups formally affiliated with NCTM that work in
partnership with the Council to meet mutual goals. As a professional association, the NCTM derives its strength
from the involvement of its members, who are drawn from the broad community of stakeholders interested in the
field of mathematics and mathematics education.
NATIONAL ACADEMY OF SCIENCES · NATIONAL RESEARCH COUNCIL
CENTER FOR SCIENCE, MATHEMATICS, AND ENGINEERING EDUCATION
MATHEMATICAL SCIENCES EDUCATION BOARD
The National Academy of Sciences (NAS or the Academy) is a private, nonprofit, self-perpetuating society of
distinguished scholars engaged in scientific and engineering research and dedicated to the furtherance of science
and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the
Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and
technical matters. The National Research Council (NRC or the Council) was organized by the National Academy of
Sciences in 1916 to associate the broad community of science and technology with the Academy's purposes of
furthering knowledge and advising the federal government. Functioning in accordance with general policies
determined by the Academy, the Council has become the principal operating agency of both the National Academy
of Sciences and the National Academy of Engineering in providing services to the government, the public, and the
scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of
Medicine.
The Mathematical Sciences Education Board (MSEB) was established in 1985 by the National Research
Council to maintain a national capability for assessing the status and quality of mathematics education. The MSEB
is located within the Center for Science, Mathematics, and Engineering Education (CSMEE or the Center), which
was established in 1995 to provide coordination of the NRC's education activities and reform efforts for all students
at all levels, specifically those in kindergarten through twelfth grade and in undergraduate, school-to-work, and
continuing education programs in the disciplines of science, mathematics, technology, and engineering. The Center
reports directly to the Governing Board of the NRC.
. . .
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Acknowl~ed~ments
The National Council of Teachers of Mathematics (NCTM) and the Mathematical Sciences Education Board
(MSEB) both have as their mission the improvement of mathematics education. Because algebra is one of the
cross-cutting content areas in the mathematics curriculum and a topic that is currently of much concern to the
mathematics community at large, the two organizations joined together to organize a national symposium on
algebra in May of 1997. This joint venture was unique in the relationship between the two organizations and
represents a significant step in bridging the diverse communities represented by the two organizations. The
symposium was organized by the Algebra Symposium Task Force of NCTM and a subgroup of MSEB members.
We gratefully acknowledge the National Science Foundation (NSF), whose financial support (Award
#9614977) made the symposium possible, and Texas Instruments and Casio, who provided additional funds. Any
opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do
not necessarily reflect those of the funders.
We also would like to acknowledge the staff at NCTM and at MSEB for their efforts in putting the symposium
together. In particular, Ramona Irvin from MSEB and Nancy Hawthorne from NCTM spent many hours drawing up
invitation lists, contacting participants, arranging housing, and, in general, ensuring that the details were in place for
a successful meeting. They were supported in their efforts by Catherine Bell and Colleen McGurkin from MSEB
and Kathleen Chapman and Mary Ferris from NCTM. A special thank-you goes to Marilyn Hala from NCTM for
shepherding the grant-writing process and for her help on-site during the symposium. Others who provided on-site
support were Francis (Skip) Fennell and Bradford Findell from the MSEB staff and Virginia Williams, Joan
Armistead, and Kathleen Chapman from the NCTM staff.
We are grateful to the speakers for their contributions and leadership that gave substance to the discussion and
to Mark Saul and Bill Tate for providing thoughtful pre-conference readings. Finally, we would like to thank
Francis (Skip) Fennell, James Gates, Kathleen (Kit) Johnston, and Beth Wallace from MSEB for their work in
organizing and editing these proceedings for review and publication. A special thanks goes to Bradford Findell for
his review of the mathematics in this work.
It should be noted that these proceedings have been reviewed by individuals chosen for their diverse
perspectives and technical expertise, in accordance with procedures approved by the National Research Council's
(NRC) Report Review Committee. The purpose of this independent review is to provide candid and critical
comments that will assist the NRC in making the published report as sound as possible and to ensure that the report
meets institutional standards for objectivity, evidence, and responsiveness to the study charge. The content of the
review comments and draft manuscript remains confidential to protect the integrity of the deliberative process.
v
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v!
ACKNOWLEDGMENTS
We wish to thank the following individuals for their participation in the review of this report:
Dr. Christian Hirsch, Western Michigan University
Dr. Roger Howe (NAS), Yale University
Dr. Henry O. Pollak, retired
Dr. Cathy L. Seeley, University of Texas
Ms. Bonnie Walker, Texas ASCD
While the individuals listed above have provided many constructive comments and suggestions, responsibility
for the final content of this report rests solely with the authoring committee and the NRC.
GAIL BURRILL
President
National Council of Teachers of Mathematics
JOAN FERRINI-MUNDY
Director
Mathematical Sciences Education Board
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NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS
ALGEBRA SYMPOSIUM TASK FORCE
Beverly Williams, Chair
Pulaski County Special School District
Sherwood, Arkansas
Hyman Bass*
Columbia University
New York, New York
Laurie A. Boswell
Profile School
Bethlehem, New Hampshire
Sadie C. Bragg*
Borough of Manhattan Community College
The City University of New York
New York, New York
Gail F. Burrill
University of Wisconsin
Madison, Wisconsin
*Mathematical Sciences Education Board members subgroup
vat
Leigh Childs
San Diego State University
San Diego, California
Shari Ann Wilson Coston*
Arkansas Education Renewal Consortium
Arkadelphia, Arkansas
Robert L. Devaney
Boston University
Boston, Massachusetts
Irvin E. Vance
Michigan State University
East Lansing, Michigan
Bert K. Waits
Ohio State University
Columbus, Ohio
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Preface
The National Academy of Sciences was pleased to host on May 27 and 28, 1997, a national symposium on "The
Nature and Role of Algebra in the K-14 Curriculum" jointly sponsored by the National Council of Teachers of
Mathematics (NCTM) and the National Research Council's Mathematical Sciences Education Board (MSEB).
One of the Academy' s greatest strengths lies in its ability to act as a convener. As I observed the symposium on
its last day, I heard and saw the breadth of the representation across grade levels and across states. NCTM and
MSEB clearly succeeded in their ongoing commitment to bring together thoughtful members of the mathematical
sciences community to consider important questions in mathematics education. In this case, the questions involved
the timely topic of algebra and how it should be treated in the K-14 grades.
This record of the symposium proceedings reflects the diversity of the symposium's speakers and participants.
It is rich with the shared information and perspectives of elementary-, middle-, and high-school teachers, postsec-
ondary and research mathematicians, teacher educators, mathematics education administrators, and others.
As you read the papers, presentations, and discourse of the symposium's two days here at the Academy, you
will see that the subject of the nature and role of algebra in the K-14 curriculum is difficult and complicated. The
questions that are being asked include, What do we mean by algebra and algebraic thinking? What do American
students really need to know about and be able to do with algebra? How can we better prepare K-14 teachers to
teach algebra? How can we better communicate to parents, the business community, and the general public about
the kind of algebra that is relevant and why?
As the mathematics and mathematics education communities work with these questions, the Academy, the
National Research Council, and the MSEB will continue to be active participants and partners. This is a high-stakes
matter, and it will take all of our efforts to make sure that our nation and our nation's children are mathematically
prepared for the 21 st century.
Bruce Alberts
President
National Academy of Sciences
x~
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Investigate the relationship between the derivative of a function and the key features of its graph.
Understand rates of change and their applications.
Understand the concept of a limit of a function, and be able to evaluate the limit of a function.
Use the chain rule.
Integrals are often omitted in many of the high school calculus curricula, however, in some courses the application of integrals will still apply. Pre-Calculus
(Sometimes this is offered in 11 or 12, this course tends to include early Calculus concepts.)
|
Imagine Deerfield
Give/Volunteer for Deerfield
Share this page
Function with Confidence
Calculators
The Mathematics Department requires that ALL students own a TI-84 graphics calculator. Students may purchase a graphics calculator at Hitchcock House or in most department stores.
In the Mathematics Department, the faculty encourages students to develop the ideas, skills, and attitudes that will enable them to function with confidence and intelligence in a swiftly changing world. In pursuing this goal, teachers strive to instill a sense of excitement for the concepts and aesthetic qualities of mathematics. Deerfield students learn how to solve mathematical problems with a variety of strategies, how to communicate their solutions clearly, how to work effectively on projects with their peers, and how to use technology. The department offers a variety of courses, and places students into a level of mathematics that will provide appropriate challenges and successes. For example, the department teaches three levels of Algebra II, and at the higher end of the spectrum, outstanding students may study college-level mathematics in one-on-one tutorial classes.
For entering freshmen who have been advised to take Algebra I (Math 101 or Math 102), the usual sequence of mathematics courses consists of Math 102 (or Math 101), Math 202 (or Math 201), Math 302 (or Math 301) and Math 402 (or Math 401). For entering freshmen who place out of Algebra I (Math 101 or Math 102), the usual sequence is Math 202 (or Math 201), Math 302 (or Math 301), Math 402 (or Math 401), and Math 602 (or Math 503). The department will annually guide students in the selection of a program that is appropriate in both content and pace. Accelerated and enriched courses (Math 203, 303 and 403) provide an alternative to the usual sequence and permit advancement towards AP Calculus courses (either Math 602 or 603) and beyond. Students who are very successful in Math 401 or Math 402 or higher courses are eligible to take AP Statistics.
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Cairine Wilson Secondary School
Course Outline
Principles of Mathematics
MPM1D (9)
Academic
Prerequisite: None
Description: This course enables students to develop an understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and hands-on activities. Students will investigate real-life examples to develop various representations of linear relations, and will determine the connections between the representations. They will also explore certain relationships that emerge from the measurement of three-dimensional figures and two-dimensional shapes. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
Overall Expectations:
• demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
• manipulate numerical and polynomial expressions, and solve first-degree equations.
• apply data-management techniques to investigate relationships between two variables;
• demonstrate an understanding of the characteristics of a linear relation;
• connect various representations of a linear relation.
• determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
• determine, through investigation, the properties of the slope and y-intercept of a linear relation;
• solve problems involving linear relations.
• determine, through investigation, the optimal values of various measurements;
• solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional figures;
• verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-dimensional shapes, and apply the results to solving problems.
Resources: Principles of Mathematics 9 (Nelson) $75.95
Learning Skills: The separate evaluation and reporting of the learning skills in the following five areas reflects their critical role in students' achievement of the curriculum expectations. Students will be assessed continually on the following learning skills:
Works Independently
Teamwork
Organization
Work Habits/Homework
Initiative
Self-Regulation
accepts responsibility for completing tasks, follows instructions, completes assignments on time and with care, uses time effectively
works willingly and cooperatively with others, is sensitive to the needs of others, takes responsibility in sharing the work, shows respect for others ideas and opinions
sets own individual goals and monitors progress towards achieving them
Assessment and Evaluation Policy
Insufficient
Evident
Response
Late, Missed or
Skipped Tasks
(Parents are
reminded to
contact the school for all absences)
· The student will be consulted regarding the reason
· The parent/guardian will be contacted
· A second due date will be negotiated.
· If the task is not submitted according to the negotiated second due date
deductions of 10% per day up to and including "0" may be awarded in
consultation with the School Success team which may include Department Head, Administration, and Guidance.
· Students who miss assessment tasks have presented zero evidence of learning. Based on the professional judgment of the teacher, students may be required to complete the assignment in order to meet the overall expectations of the curriculum.
· A final mark of " I " or "insufficient evidence" is acceptable for grade 9 and 10 course
Academic Integrity
· Fraudulent work is of no value and provides zero evident of learning.
· Intentional academic fraud is a disciplinary issue and will incur consequences which may include suspension and mark reduction.
· Teachers will take into account mitigating circumstances when dealing with academic fraud.
· Students will be given an additional opportunity to demonstrate achievement when in the teacher's professional judgment there is not sufficient evidence that the student has met overall course expectations.
· Fraudulent material will be documented and archived.
· The parent/guardian will be contacted.
· Students who commit intentional academic fraud will forfeit the possibility of winning subject awards.
· All students in grade 9 will be required to attend academic integrity workshops at the beginning of each school year.
Extra Help:The staff of CW is committed to the success of all students. Students are strongly encouraged to seek extra help from the teacher both in and out of the classroom. The Green room is available after school Tuesday, Wednesday and Thursday.
Communication: Please feel free to contact me at the school ,613-824-4411, if you have any questions or concerns. My Voicemail extension is ________ and my email address is _______________________
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Frames for Undergraduates
9780821842126
ISBN:
0821842129
Edition: 1 Pub Date: 2008 Publisher: American Mathematical Society
Summary: "Frames for Undergraduates is an undergraduate-level introduction to the theory of frames in a Hilbert space. This book can serve as a text for a special-topics course in frame theory, but it could also be used to teach a second semester of liner algebra, using frames as an application of the theoretical concepts. It can also provide a complete and helpful resource for students doing undergraduate research projects u...sing frames." "The early chapters contain the topics from linear algebra that students need to know in order to read the rest of the book. The later chapters are devoted to advanced topics, which allow students with more experience to study more intricate types of frames. Toward that end, a Student Presentation section gives detailed proofs of fairly technical results with the intention that a student could work out these proofs independently and prepare a presentation to a class or research group. The authors have also presented some stories in the Anecdotes section about how this material has motivated and influenced their students."--BOOK JACKET.[read more]
Ships From:Salem, ORShipping:Standard, ExpeditedComments:Has minor wear and/or markings. SKU:9780821842126-3-0-3 Orders ship the same or next business day... [more] A frame in a finite-dimensional inner-produ... [more] [[ adaptability to existing conditions allows frames to be used in applied settings including signal processing, imaging, sampling, and cryptography. The study of frames, particularly in finite dimensions, begins with exactly the topics from an undergraduate linear algebra course. This makes the topics particularly accessible to undergraduate students, yet the theory contains deep unsolved problems. This book can be used as a resource for an REU or for a topics course about frames. It is also a suitable textbook for a second linear algebra course, using frames as a thematic example to demonstrate and explore the new material. The theory of frames is increasingly broad with widespread applications. "Frames for Undergraduates" introduces students to this vibrant and important area of mathematics.[less]
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Chapter 2 Homework
Try to do enough problems from a section until you're
comfortable doing problems of the kind found in that section.
Should you have trouble with the problems listed here,
any of the odd numbered problems around these would be good to try--the answers
are in the back of the book and you have a student solutions
manual that shows you how to do them.
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When students truly understand the mathematical concepts, it's magic. Students who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement of the mathematical world.
That—from the textbook, to the eManipulative activities, to the online problem-solving tools and the resource-rich website—work in harmony to help achieve this goal. This edition can also be accompanied with WileyPlus, an online teaching and learning environment that integrates the entire digital textbook with the most effective resources to fit every learning style.
for Mathematics for Elementary Teachers: A Contemporary Approach, 9th Edition. Learn more at WileyPLUS.com
Revised and Enriched Exercise Sets: This where necessary, to assure that they are closely aligned with and provide complete coverage to the section material. Exercises in Exercise/Problem Sets A and B are arranged in matched pairs, but problems are not. Answers are provided for all Set A exercises/problems. Answers are provided for all Set B exercises/problems in the Instructor Resource Manual.
Section Rearranged: Section 2.4 from the 8th edition has been moved to Sec 9.3 in this edition to enrich the coverage of algebra.
Chapter Revisions: Chapter 12 has been substantially revised. Sections 12.1 and 12.2 have been organized to more faithfully represent the first three van Hiele levels. In this way, students will be able to pass through the levels in a more meaningful fashion so that they will get a strong feeling about how their students will view geometry at various grade levels.
Algebraic Reasoning: To further enrich the coverage of algebra, Algebraic Reasoning margins notes have been strategically placed throughout the book to help students see how what they are studying is connected to algebra.
Check for Understanding: To help students be more active when learning the material, Check for Understanding callouts lead students to Part A exercises that are relevant to the subsection they just finished studying.
Analyzing Student Thinking: Problems have been added to the end of the Part B problems. These problems pose questions that students may face when they teach. Many of the problems that were marked with fountain pen icons at the end of the Exercise/Problems sets have been incorporated into the Analyzing Student Thinking problems in this edition.
Author Walk-Throughs: These are audio vignettes that precede each chapter and each section. These brief vignettes help students hear about points of major emphasis in each chapter/section so that their study can be more focused.
Revised Children's Literature & Reflections from Research: These margins have been revised and refreshed.
Revised Website: Nearly all of the Problems for Writing/Discussion that preceded the Chapter Tests in the 8th edition now appear on our Website.
Problem-Solving Emphasis: Features the largest collection of problems (over 3,000!), worked examples, and problem-solving strategies in any text of its kind.
Integrated Technology: Technology and content are integrated throughout the text in a meaningful way. The technology includes WileyPLUS, activities from the expanded eManipulative activities, spreadsheet activities, Geometer's Sketchpad activities, and calculator activities using a graphics calculator and Math Explorer.
Extensive Geometry Coverage: Comprehensive, five-chapter treatment of geometry based on the van Hiele model provides students with the necessary, and often neglected, background in geometry.
Appropriate Topical Sequence: Moves from the concrete to the pictorial to the abstract, reflecting the way math is generally taught in elementary schools.
NCTM Standards: These are referenced throughout in the margin and reviewed at the beginning of each chapter, giving future teachers a good idea of the standards they will be responsible for and the skills that their students will be tested on.
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Math 366 - Numerical Analysis
Spring Semester, 1997-98
One of the best definitions of numerical analysis appeared several years
ago in SIAM News. In a commentary, Lloyd N. Trefethen defined
numerical analysis as "the study of algorithms for the problems of
continuous mathematics." Thus, in view of this definition, a
numerical analyst is a mathematician who develops, analyzes, and
evaluates algorithms for obtaining (approximate) solutions to
mathematical problems.
Numerical analysis has evolved to the point where it is regarded as a
branch of mathematics in its own right, but it has strong roots and ties
to the applications of mathematics and the development of computer science
and technology. It involves applying the power of mathematics and the
power of the computer to solving quantitative problems in science and
engineering.
class attendance and participation; attendance at three or more department colloquia and other designated special events (10%).
***There will be no make-up exams, and late work will not be accepted.***
DAILY READINGS:
Read the textbook! Assigned readings should be done before class so that you will have some familiarity with the new material when it is discussed in class.
Working problems is essential for an understanding of the material, and there is an adequate supply of problems in the textbook.
TEAM HOMEWORK:
Homework will be assigned, collected, and graded. It is due at the beginning of class. As noted under GRADING POLICY, late homework will not be accepted.
Assignments may include material that will not be discussed in class. You are expected to learn this material on your own and to make use of the resources available to you to complete the assignments.
Homework will be completed by teams of three people with each team producing a single write-up.
Homework must be written neatly in standard English with complete sentences using standard 8-1/2 by 11 paper (no ragged edges) in a style appropriate to the subject. Express yourself clearly and concisely. Do not write to me. Assume instead that you are writing to other students in the class.
Grading will be based both on mathematical content and on the quality of your write-up. NEATNESS COUNTS! Show all work necessary to justify your solutions. Answers alone are not sufficient.
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If you missed more than two of these problems, you should brush up on basic algebra. These skills are needed in several of the following sections of this self-assessment, so it would be wise to do that study before moving on to the next sections. In addition, most learning resources covering the content in sections 2-13 below presume mastery of basic algebra.
Resources for Brushing Up On Basic Algebra
GSEHD's doctoral level quantitative research methods and statistics courses require only the equivalent of the first semester of what is normally called Algebra I. You need to know what equations represent and how to re-write equations to solve for any given unknown.
Arithmetic and Algebra Again Brita Immergut and Jean Burr Smith; McGraw-Hill, 1994; $15. This book provides good explanations of pre-algebra math, the basics of algebra, some descriptive statistics, and graphs. There are lots of problems and an answer key.
Practical Algebra: A Self-Teaching Guide, Second Edition Peter Selby & Steve Slavin; John Wiley, 1991; $20. This book also covers pre-algebra math, the basics of algebra, some descriptive statistics, graphs, and probability. There are lots of problems and an answer key.
algebrahelp.com This is a good online source, providing in-depth lessons with problem sets and answers. Site also contains an interactive equation solver.
Mathematics Skills Review Series #6: Review of Algebra This is an instructional Web site with generally clear explanations, good illustrations, and lots of exercises with answers. It was developed by six universities with federal funding. (Note: After selecting a module, you have to click on the small yellow cross at the bottom to get to the content links.)
MathPrep.com This Web site offers several self-paced online mathematics courses. The Elementary Algebra course covers pre-algebra, all the algebra you will need in the GSEHD doctoral courses, and graphs. The charge is $70. Problems are accompanied by answers and usually examples showing one way to arrive at the correct answer.
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Algebra
posted on: 10 Dec, 2011 | updated on: 22 Sep, 2012
Algebra is an important branch of mathematics in which we deal with several types of problems. These problems may either related to constants or variables. It is an interesting part of Math, and in this we mainly concentrate on "how to solve Numbers"? When we solve any algebra problem we solve numbers only. There are several games using which you can easily learn the concept of algebra like kids have a great fun when they solve the puzzles, play some computer games by running, finding secret doors, etc.
When we deal with algebra we mainly focus on equations and expression. These two terms can be defined as the heart of the algebra, as whenever we solve any problem we have to solve different equations. Equations are mathematical statements, which show the equality of two different numbers or expressions. Using algebra we solve the equation problems. Algebra is much broader than elementary algebra, in algebra we use different type of rules and operations, and perform all operations. In this, the variable symbol that represents numbers and expressions are mathematical termed as variables, numbers or both. On the different side, expressions are the, mathematical phrases, which don't use equal to symbol as it doesn't show any equality.
Apart from algebra, there are several different branches of mathematics like Geometry, trigonometry, calculus, etc. To understand all these branches properly your base must be strong, which can only develop with help of algebra. So, learn algebra properly and become master of math.
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Mathematics
MATHEMATICS AT FRIENDS ACADEMY is a comprehensive program which combines the essential elements of computation, concepts, and application. Through our curriculum we facilitate learning by providing relevant, multidimensional learning experiences with special emphasis on the processes of mathematics. Mathematics at Friends Academy is characterized by small class sizes and personal attention from the faculty, including advisement of students on course scheduling and career opportunities.
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UIL ACADEMIC CONTESTS
The University Interscholastic League offers a wide variety of academic contests for high school students, encompassing many elements of required high school coursework. These contests build upon the academic skills developed in the classroom and offer students an opportunity to stretch their talents above and beyond those requirements. The complete rules for each contest are contained in the UIL Constitution and Contest Rules, which is updated every year and posted on the UIL web site at:
Accounting Sponsor: Mrs. Bainum (mbainum@canyonisd.net)
Maybe you're on your way to becoming a CPA or you just really know how to take care of money. Make a stop at this contest and pick up a few skills in bookkeeping, balancing and banking before you take one of the Big Four accounting firms by storm.
The contest focuses on the elementary principles and practices of accounting for sole proprietorship, partnerships and corporations, and includes bookkeeping terminology, the work sheet with adjustments, income statement, balance sheet, trial balance, account classification, journalizing, posting, bank reconciliation, payroll and other items related to the basic accounting cycle.
Calculator Applications Sponsor: Mrs. Taylor (mtaylor@canyonisd.net)
Calculate this: Add your math skills to a college application, standardized test or resume, and success might just be the result. Math is power in today's job market, so multiply your potential by trying out this problem-solving contest.
The contest includes calculations involving addition, subtraction, multiplication, division, roots, powers, exponentiation, logarithms, trigonometric functions and inverse trigonometric functions. In addition to straightforward calculation problems, the contest includes geometric and stated problems similar to those found in algebra, geometry and trigonometry textbooks, previous contests and League materials related to the contest.
Computer Applications Sponsor: Mr. Culbert (lculbert@canyonisd.net)
For the 'tech' generation: Become technologically savvy while testing your word processing, database and spreadsheet skills. You'll become familiar with the finer points of computer skills such as formatting copy, editing, creating charts and integrating applications.
Computer Science Sponsor: Mr. Culbert (lculbert@canyonisd.net)
The Computer Science contest challenges students to study a broad range of areas in computer science and has both an individual and a team component. Competition consists of a 45-minute written exam for both components, along with a two-hour hands-on programming contest for teams.
At all levels of competition, individual places are determined solely by written exam scores. All contestants compete for individual honors at all levels of competition. Individuals placing first, second and third advance to the next level. For the team component, the team's top three scores on the written exam are added to its score on the hands-on contest to produce an overall team score. First-place teams advance to the next level of competition.
Current Issues & Events Sponsor: Mr. Cochran (jcochran@canyonisd.net)
You'll go around the world in 40 multiple-choice questions as you test your knowledge on current state, national and global events. Watching news shows will pay off when you answer the essay question at the end and take a closer look at one current event.
The contest focuses on a basic knowledge of current state, national and world events and issues. The contest consists of 40 multiple-choice questions and an essay that challenges students to understand not just what is happening in the world today, but why and how it's happening and what it means to us as citizens of the United States.
CX Debate Sponsor: Ms. Suto (ssuto@canyonisd.net)
If you've never shied away from an argument and you have a zest for winning, give Cross-Examination Debate a try. As part of a two-person team, you will prepare your stance on a particular policy in advance and then face an opposing team in competition. You'll have to think on your feet to defend your ideas.
Cross-Examination Debate trains students to analyze a problem, conduct thorough and relevant research, and utilize principles of argumentation and advocacy in presenting the most effective case for or against a given proposition. Debate provides invaluable training in critical thinking, quick responses, defending worthy ideas and attacking invalid ideas. It teaches students to tolerate other points of view. Debate exists only in democratic societies, and no democratic society can exist without debate.
Editorial Writing Sponsor: Mrs. Smith (llsmith@canyonisd.net)
Have you ever won a medal for simply writing down your opinion in an organized way? In editorial writing, you'll take a stand on a controversial school issue and back up your stance with facts and examples.
This contest teaches students to read critically, to digest and prioritize information quickly, and to write clearly, accurately and succinctly. Emphasis is placed on mechanical and stylistic precision, lead writing, use of direct and indirect quotes, news judgment, and the ability to think deeply, to compare and contrast and to argue or defend a point of view persuasively.
Feature Writing Sponsor: Mrs. Smith (llsmith@canyonisd.net)
If you've got a knack for developing a story, this contest is for you. You'll be provided with the facts and quotes you need, and then it's up to you to piece together a journalistic feature story your readers will remember.
The Feature Writing Contest teaches students to read critically, to digest and prioritize information quickly, and to write clearly, accurately and succinctly. Emphasis is placed on the same writing skills as in other UIL journalism contests, as well as the ability to write descriptively.
Headline Writing Sponsor: Mrs. Smith (llsmith@canyonisd.net)
Put the finishing touches on the news as you decide what's most important about six news stories and top them off with headlines. The challenge is to be creative in your word choice and adhere to the word and line counts as you write tomorrow's headlines.
The contest teaches students to read critically, to digest and prioritize information quickly, and to write clearly, accurately and succinctly. Emphasis is placed on the ability to discern key facts and to write with flair and style in order to tell and sell a story.
Informative Speaking Sponsor: Ms. Suto (ssuto@canyonisd.net)
This contest is all about watching the clock and knowing your material. You'll draw a current event and have 30 minutes to comb through files you've collected throughout the year. Then you'll present a speech that informs your audience on all aspects of the current event you've researched.
The purpose of informative speaking is to stimulate an active interest in current affairs at the state, national and international levels, and to teach the student to present extemporaneously in a clear and impartial manner the facts about a subject as they appear in the best available sources of information. This contest is an exercise in clear thinking and informing the public on the issues and concerns of the American people. The objective is to present information in an interesting way, and an attempt should not be made to change the listener's mind beyond presenting the information.
Lincoln-Douglas Debate Sponsor: Ms. Suto (ssuto@canyonisd.net)
In this one-on-one values debate, you'll prepare to argue for and against a given resolution. After researching the topic in advance, it will be up to you to make arguments that defend your point of view and debunk invalid claims from your opponent.
Lincoln-Douglas debate provides excellent training for development of skills in argumentation, persuasion, research and audience analysis. Through this contest, students are encouraged to develop a direct and communicative style of delivery. Lincoln-Douglas debate is a one-on-one argumentation in which debaters attempt to convince the judge of the acceptability of their side of a proposition. One debater will argue the affirmative side of the resolution and the other will argue the negative side of the resolution in a given round.
You'll need a critical eye as you scan through literary history. You'll analyze literary from a provided reading list as well as literary passages not on the list. A short essay serves as the tiebreaker that could put you over the top.
The contest requires knowledge of literary history and of critical terms, and ability in literary criticism. Students are required to select the best answers involving judgment in literary criticism and to analyze literary passages from both the reading list and other sources. A tiebreaker is required in which the student must write a short essay dealing with a specified topic about a selected literary passage.
Mathematics Sponsor: Mrs. Taylor (mtaylor@canyonisd.net)
Algebra, geometry, pre-calculus, oh my! Come armed for this test with your knowledge and understanding of a variety of mathematical subjects such as geometry and trigonometry as you compete against your peers.
This 40-minute, 60-question contest is designed to test knowledge and understanding in the areas of Algebra I and II, Geometry, Trigonometry, Math Analysis, Analytic Geometry, Pre-Calculus and Elementary Calculus.
News Writing Sponsor: Mrs. Smith (llsmith@canyonisd.net)
In this contest, you decide what's fit to print as you make your way through a set of facts and quotes, and pick out what's important. You'll work on deadline for the newspaper as you create a cohesive story that inquiring minds have a right to know.
The News Writing Contest teaches students to read critically, to digest and prioritize information quickly, and to write clearly, accurately and succinctly. Emphasis is placed on mechanical and stylistic precision, lead writing, use of direct and indirect quotes, and news judgment.
Number Sense Sponsor: Mrs. Taylor (mtaylor@canyonisd.net)
Ten minutes is all it takes to find out if you have good number sense. You'll work with your coach and team to develop and practice shortcuts to solve the mental math test and still beat the clock. Make sense?
This 80-question mental math contest covers all high school mathematics curricula. All answers must be derived without using scratch paper or a calculator.
One-Act Play Sponsor: Mr. Yirak (dyirak@canyonisd.net)
Before you make pack up your bags and shuffle off to Broadway, try out the League's One-Act Play contest. You'll have the chance to work with other actors and people interested in technical theatre at your school to produce a theatrical production. You'll get a chance to take your show on the road and compete against other schools and you might just make it to the state competition. Many of Texas' best theatre and film professionals participated in this contest while in high school.
The aims of the One-Act Play Contest are to satisfy the competitive, artistic spirit with friendly rivalry among schools, emphasizing high quality performance in this creative art; to foster appreciation of good acting, good directing and good theatre; to promote interest in that art form most readily usable in leisure time during adult life; to learn to lose or win graciously; and to increase the number of schools which have adopted theatre arts as an academic subject in school curricula.
Persuasive Speaking Sponsor: Ms. Suto (ssuto@canyonisd.net)
Similar to informative speaking, in this contest you have 30 minutes to review your research files on a particular current event and come to a conclusion to argue about that topic. The goal of your speech is not just to present relevant information, but to convince your audience that your position is solid.
This contest trains students to analyze a current issue, determine a point of view, and organize and deliver a speech that seeks to persuade listeners. The objective is to reinforce the views of listeners who already believe as the speaker does, but even more so, to bring those of neutral or opposing views around to the speaker's beliefs or proposed course of action. This contest should especially appeal to those who have a strong argumentative urge and who wish to advocate reforms or outline solutions to current problems.
Poetry Interpretation Sponsor: Ms. Suto (ssuto@canyonisd.net)
In poetry interpretation, you'll choose a selection that fits in the given category to present to an audience. This contest emphasizes literary analysis through expressive oral reading.
The purpose of this contest is to encourage the student to understand, experience and share poetry through the art of oral interpretation. The goals of this contest are to encourage the contestant's exploration of a variety of literary selections, and to enhance the performer's and audience's appreciation of literature through the performer's interpretation of the work.
Prose Interpretation Sponsor: Ms. Suto (ssuto@canyonisd.net)
Those with a flare for expressive oral reading have a chance to combine their passions in this event. You'll select a piece of prose in a given category, then carefully explore the art of expressing it orally before an audience.
This contest encourages the student to understand, experience, and share prose works through the art of oral interpretation. It encourages the contestant's exploration of a variety of literary selections and enhances the performer's and audience's appreciation of literature through the performer's interpretation of the work.
Ready Writing Sponsor: Mrs. Hale (hhale@canyonisd.net)
Ready, set, write! If you like to make your own path, this contest is for you. A short prompt will provide the inspiration for your creative ideas as you explore a topic or prove a point.
Students write expository compositions that attempt to explain, prove or explore a topic in a balanced way, allowing the argument and the evidence given to be the deciding factor in the paper. Students are given a choice between two prompts, each an excerpt from literature, publications or speeches. The essay is judged on interest, organization and style.
Science Sponsor: Ms. Wieck (awieck@canyonisd.net)
Forget just memorizing facts, because the science contest is all about the importance of experiments and scientific discoveries. Your knowledge of biology, chemistry and physics will help you select the correct answers on this 60-question multiple-choice test. Individual awards are given in each subject area, so even students who have not yet taken all the science courses can excel!
The Science Contest challenges students to read widely in biology, chemistry and physics, to understand the significance of experiments rather than to recall obscure details, and to be alert to new discoveries and information in the areas of science. It is designed to help students gain an understanding of the basic principles as well as knowledge of the history and philosophy of science, and to foster a sense of enthusiasm about science and how it affects our daily lives.
Social Studies Sponsor: Mr. Cochran (jcochran@canyonisd.net)
If your interest lies in movements, wars, history and politics, this contest will give you more than enough material to explore. The contest requires you to apply your understanding of history and culture through multiple-choice questions and an essay.
The Social Studies Contest requires students to expand and apply their knowledge of governmental systems; historical trends, movements and eras; and the physical setting of the earth, particularly as it applies to cultural environments. Each year the contest focuses on a selected topic area, and a reading list is provided.
Spelling & Vocabulary Sponsor:
Whether you've already aced the SAT verbal section or you could use some extra practice, this contest keeps you focused on the details. By the end, you may be correcting your teachers' spelling and using words your coach has never heard.
Spelling & Vocabulary promotes precise and effective use of words. The three-part contest consists of multiple-choice questions covering proofreading and vocabulary, and words that are written from dictation. The vocabulary-building and spelling components of the contest are important complements to the high school academic curriculum and are indicative of vocabulary words contained on standardized tests such as SAT, PSAT and ACT.
Texas Interscholastic League Foundation Scholarships
The Texas Interscholastic League Foundation has funded over $21 million in scholarships to more than 15,000 students since 1954. The TILF was chartered in 1959 and will soon be celebrating its 50th Anniversary. For the past several years the foundation has disbursed over $1 million each year to students who have earned eligibility by competing in the UIL Academic State Meet. Students may apply for TILF scholarships the year they are graduating from high school, but may earn eligibility any of their high school years. Scholarship values range from $500 for one year only, up to $15,200 payable over four years. Applications are accepted April 1st through the Tuesday following the UIL Academic State Meet. Current and updated web site information should be available beginning March 1 of each year. Approximately 50% of applicants receive a scholarship.
The purpose of the TLIF program is to try and help as many students as possible, who may not otherwise have the opportunity to do so, attend a college or university in Texas. Our hope and mission for the TILF program is that each qualified student who applies receives some financial support. However, we have not yet reached that goal. Therefore, donor foundations, organizations and individuals must make difficult funding decisions. Recipients are selected on a number of factors, which include but are not limited to, UIL involvement and placement, ACT/SAT scores (78% of those selected for the 2006-07 academic year scored above 1200 on the SAT), class ranking, high school GPA, letters of recommendation, and the student's financial situation, since the majority of these scholarships are need based.
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Greetings, I am a freshman in college and I'm struggling with my homework assignments. One of my frustrations is understanding negative and positive worksheets 6th grade; Could anyone out there on the ethernet help me with understanding what it is all about? I want to complete this yesterday! Thanks for your time.
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Registered: 24.10.2003
From: Where the trout streams flow and the air is nice
Posted: Tuesday 30th of Dec 16:28
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Sharpen your skills and prepare for your precalculus exam with a wealth of essential facts in a quick-and-easy Q&A format! Get the question-and-answer practice you need with McGraw-Hill's 500 College Precalculus Questions. Organized for easy reference and intensive practice, the questions cover all essential precalculus topics and include detailed answer explanations. The 500 practice questions are similar to course exam questions so you will know what to expect on test day. Each question includes a fully detailed answer that puts the subject i... MOREn context. This additional practice helps you build your knowledge, strengthen test-taking skills, and build confidence. From ethical theory to epistemology, this book covers the key topics in precalculus. Prepare for exam day with: 500 essential precalculus questions and answers organized by subject Detailed answers that provide important context for studying Content that follows the current college 101 course curriculum
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Short description is part of our range of Key Stage 3 (KS3) maths eBooks that are fully aligned with the UK Governments national curriculum.
This eBook is part of our range of Key Stage 3 (KS3) maths eBooks that are fully aligned with the UK Governments national curriculum.
Our Key Stage 3 (KS3) maths eBooks comprise three principle sections. These are, notably: • Number and Algebra • Geometry and Measures • Handling Data In addition, there exists a Publications Guide. Our mathsNumber Patterns and Sequences is a module within the Number and Algebra principle section our Key Stage 3 (KS3) publications. It is one module out of a total of seven modules in that principle section, the others being: • Factors, Prime Numbers and Directed Numbers • Fractions, Percentages and Ratio • Decimal • Indices and Standard Index Form • Algebra • Number Patterns and Sequences • Graphs (Less)
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Summary: Check your work and reinforce your understanding with this manual, which contains complete solutions for all odd-numbered exercises in the text. You will also find problem-solving strategies plus additional algebra steps and review for selected problems26.35
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Students proficient in Mathematical Modeling should demonstrate the ability to
create mathematical models of empirical or theoretical phenomena in domains such as the physical, natural, or social sciences;
create variables and other abstractions to solve college-level mathematical problems in conjunction with previously-learned fundamental mathematical skills such as algebra;
draw inferences from models using college-level mathematical techniques including problem solving, quantitative reasoning, and exploration using multiple representations such as equations, tables, and graphs.
A passing grade in an approved course is required to show proficiency in mathematical modeling under the General Education curriculum.
Course Characteristics
1. Mathematical modeling courses
a. are mathematics courses that either are required for students in the natural and mathematical sciences or address problems through mathematical models;
b. emphasize mathematical rigor and abstraction, fundamental mathematical skills, and college-level mathematical concepts and techniques;
c. teach how to develop mathematical models and draw inferences from them;
d. include a full semester or equivalent of frequent and regular assignments that provide practice in mathematical modeling and mathematical techniques. Problems providing modeling practice
i. are phrased with limited use of mathematical notation and symbols;
ii. require a formulation step on the part of the student;
iii. require college-level mathematical techniques leading from the formulation to the conclusion;
iv. have a conclusion that involves discovery or interpretation.
2. Courses approved for the Mathematical Modeling requirement must demonstrate and provide a system for consistency in instruction and in assessment of student achievement.
3. Courses approved for the mathematical modeling requirement should engage students with mathematical concepts and techniques that prepare them for a variety of possible future courses and degrees.
4. A course used to satisfy the Mathematical Modeling Foundations requirement may not double-count toward the Breadth of Inquiry Natural and Mathematical Sciences requirement.
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AcademicsCalculus
Course Outline: The course will follow
chapters 1 - 5 and 8 of Himonas and Howard's "Calculus:
Ideas and Applications" (Wiley). After this we
will briefly consider two more advanced topics depending on
the interests of the class (possibilities include multivariable
calculus, differential equations and power series) The
program will follow a strict schedule that includes drill
and practice routines for developing a familiarity with the
common tools of calculus alongside discussions surrounding
the concepts behind the subject. Thus assignments from the
text will prompt class discussion surrounding the concepts,
while daily problem sets will reinforce a command of the material.
This is a four-credit course and there is much to learn. Please
keep in mind that this course requires a daily commitment.
The best way to successfully learn the material is to stay
ahead of the game and work through the assignments before
final class discussion.
Homeworks: Bold numbered problems in the homework
assignment section below are to be handed in for grading.
Plaintext numbered problems are more drill and practice material
that should be worked through to keep up with course material.
By 4pm each Friday you are expected to hand in your solutions
to the appropriate bold numbered questions (exactly which
questions these are will be announced on the Wednesday beforehand).
You may wish to discuss any assignments that you find difficult.
|
AcademicsLinear Algebra
Course Outline: Linear Algebra is
important for its remarkable demonstration of abstraction
and idealization on the one hand, and for its applications
to many branches of math and science on the other. We will
focus our study on n-dimensional real space, considering notions
such as systems of linear equations, spanning sets, linear
independence and the matrix representations of linear transformations.
The course will follow the first seven chapters of Elementary
Linear Algebra (5th ed.) by Larson, Edwards and Falvo
(the 4th edition by Larson and Edwards is essentially the
same, but if you use this edition you will have check the
numbering of the homework assignments).
The final two weeks will be given over to students to pursue
a topic of interest further. Possibilities for this part of
the course include connections to the Differential Equations
course, the matrix representations of symmetries, consideration
of complex or finite spaces, and others.
Prerequisite: EMLS or equivalent
Grades: Your grade will be calculated as follows:
Final exam 40%, and 20% each for in-class quizzes, weekly
homeworks and project work. Class participation and
prompt submission of homework are expected. Your overall
grade may move up or down a small amount due to these factors.
Homeworks: Bold numbered problems in the Assignment
Responsibility section below are to be handed in for grading.
Plaintext numbered problems are more drill and practice material
that should be worked through to keep up with course material.
By 4pm each Friday you are expected to hand in your
solutions to the appropriate bold numbered questions (exactly
which questions these are will be announced during the previous
week). You may wish to discuss any assignments that
you find difficult with me or the math tutor.
Office hours: TBA
Tutoring: Julie Shumway (jshumway@marlboro.edu)
Assignment Responsibility. The following question
numbers are from the 5th edition of Elementary Linear Algebra.
If you have the 4th edition then you should check the copy
on the reserve shelf in the library to make sure that you
are attempting the right questions. The chapter material
is the same, except that Section 3.5 in the 5th edition is
Section 3.4 in the 4th.
|
Upon successful completion of the course, the student will be able to:
1. Use the properties of real numbers to simplify and evaluate expressions.
2. Solve linear equations and inequalities.
3. Use and transform formulas and functions.
4. Graph linear equations and inequalities in one and two variables .
5. Write the three forms of the equation of a line.
6. Solve systems of linear equations by graphing, substitution, and addition.
7. Apply the laws of exponents and use scientific notation.
8. Factor and perform operations with polynomials.
9. Solve quadratic equations by three methods: factoring, completing the square ,
and using the Quadratic Formula.
10. Graph quadratic equations.
11. Perform operations with rational algebraic expressions , and solve rational
equations.
12. Simplify and perform operations with radical expressions and rational
exponents.
13. Use exponential and logarithmic functions and the properties of logarithms.
14. Solve word problems using one or more of the above skills.
Course Requirements
Students are expected to attend all scheduled classes, do the homework assigned
each day for
the next class, take tests, and be active participants in the class.
Algebra for College Students, Mark Dugopolski, 5th edition, McGraw Hill, 2009
If purchased in the SMCC bookstore, the text will be packaged with the Student
Solutions
Manual and a MathZone Access card.
Scientific calculator (required)
Non-discrimination and Disability Statements:
Southern Maine Community College is an equal opportunity/affirmative action
institution and
employer. For more information, please call 207-741-5798.
If you have a disabling condition and wish to request accommodations in order to
have reasonable
access to the programs and services offered by SMCC, you must
register with the disability services
coordinator, Mark Krogman, who can be
reached at 741-5629.
(TTD 207-741-5667) Further information about services for students with
disabilities and the
accommodation process is available upon request at this
number.
Course Evaluation: Students may evaluate the course online and anonymously by
going to
"Resources for Current Students" at the SMCC homepage and selecting
"Evaluate Your Courses."
The online course evaluation is available to students
two weeks prior to the end date of the course.
Students cannot see a course
grade online until the online course evaluation is completed.
Grading Policy: Test will count 50% towards your final grade. Homework and
quizzes will count as
the remaining 50% of the final grade.
Homework Policy: Homework will be assigned each class and will be due on the
date specified on
the assignment calendar. Do not list answers only, but show
work that is required for the solution.
Use the student solution manual as a
guide on how I would like you to show your work. Homework
will be graded as a
10-point quiz grade. The grade will be based on your performance and
completeness of the assignment. Homework submitted late will be assessed a
deduction of 2 points
for each calendar day late.
Quiz/Test Makeup Policy: Quizzes may not be made up. You must be in class the
day of a quiz to
take the quiz and get credit for it. If you are absent the day
of a test, you have 1 calendar week to
make up the test. After the week has
elapsed, a grade of zero (0) will be assigned for the test.
Attendance Policy: Attendance in class is critical to your success in this
course and is mandatory.
Three consecutive absences will result in an automatic
failure (AF), and 5 cumulative absences will
result in an automatic failure (AF). Attendance is defined
as being in class for the whole class. If you
are more than 10 minutes late or need to leave early for any reason, you will be
counted as absent for
the class. There is no such thing as an excused absence. Make your doctor
appointments and
schedule your other commitments when you are not scheduled for class.
Tutoring Service and Extra Help: This is not an easy course and we will move
very rapidly. You
must keep up to be successful. I will be available to help you by appointment.
There are also math
tutors available free of charge and at your convenience in the Academic
Achievement Center located
on the second floor of the Campus Center. I strongly suggest visiting them early
and often to keep up
and make sure you comprehend the skills and concepts we will be studying.
Required Course Topics, MAT 108:
CHAPTER 1 THE REAL NUMBERS
all sections
Sets and The Real Numbers
Operations on the Set of Real Numbers
Evaluating Expressions
Properties of the Real Numbers and Using the Properties
|
Problem Solving George Polya
Suggestions For
Problem Solving
(from Mathematician
George Polya's book:
"How To Solve It", 1945)
Mr. Dave Clausen
La Cañada High School
How To Solve It
George Polya has four steps for solving
problems:
– 1. Understand The Problem
– 2. Devise A Plan
– 3. Carry Out The Plan
– 4. Look Back
5/30/2012 Mr. Dave Clausen 2
Understand The Problem
Is it possible to do this?
Can I verbalize what I need to do?
5/30/2012 Mr. Dave Clausen 3
Devise A Plan
Have I seen this before?
Have I seen it in a slightly different form?
Do I know a related problem?
Here is a problem related to mine that is
solved. Can I use it?
Can I restate this problem?
If I can't solve this problem, can I first
solve some related problem?
Can I solve part of the problem?
5/30/2012 Mr. Dave Clausen 4
Carry Out The Plan
Carry out the plan, checking each step as
you work to see if it makes sense.
5/30/2012 Mr. Dave Clausen 5
Look Back
Is the result what I expected?
Can I get this same result in a different
way?
Can I use this result in some other problem?
Can I use my method in a different
problem?
5/30/2012 Mr. Dave Clausen 6
|
@article {MATHEDUC.05878566,
author = {McCartney, Mark},
title = {Calculating Lyapunov exponents: applying products and evaluating integrals.},
year = {2010},
journal = {Teaching Mathematics and its Applications},
volume = {29},
number = {4},
issn = {0268-3679},
pages = {208-215},
publisher = {Oxford University Press, Oxford},
doi = {10.1093/teamat/hrq012},
abstract = {Summary: Two common examples of one-dimensional maps (the tent map and the logistic map) are generalized to cases where they have more than one control parameter. In the case of the tent map, this still allows the global Lyapunov exponent to be found analytically, and permits various properties of the resulting global Lyapunov exponents to be investigated using elementary calculus and the evaluation of certain products. A number of classroom exercises are given. (ERIC)},
msc2010 = {I95xx},
identifier = {2011c.00704},
}
|
Thanks
They are both fun courses. I would choose Math Modeling hands down. It seems more applicable to the Physicist.
Linear Algebra is good too. If you want to get in to Quantum, eigenvectors and values should be covered there. If you do take Lin Alg, don't forget to ask your teacher where in life eigenvectors/values are used. They rarely know!
To this day, whenever I am in a bank, I am able to prove that the average wait time is minimized by one queue that breaks into four tellers as opposed to four queues that feed each teller!
Hi, dont know anything about modelling, hated it in first year. The second year Linear Algebra isnt too different from first year but it starts to make more sense. I never understood what the kernel and image were untill we used matrices for differentiation so things kind of come together a bit more. Im doin the maths/phys joint but yeah for quantum physics linear algebra seems pretty important things like you use operators on eigenfunctions and get energy eigenvalues and stuff or if you have two wavefunctions that are eigenfunctions then any linear combination is solution aswell and using that schmidt orthoganilzation and stuff. QM is still a physics module though and the linear algebra part necessary was just about a page of notes so if you dont want to do it you'll know enough from first yearWell I hated abstract but liked the other two, and I was all set to do Linear before I everyone from the years above kept saying it had a high fail rate and was really hard
I also did the math/phys joint course (graduating in two weeks!) and I didn't do mathematical modelling beyond first year but the reports from others in my class were that it was an enjoyable module, athough that was when Alexei Pokrovskii (RIP) taught it. Not sure who's teaching it now or how much the course has changed since then.
I personally enjoyed linear algebra a lot more in second year than first year but I always preferred maths modules that had lots of proofs and stuff. Whereas a lot of the people in single honours physics (and even some in maths science!) hated that kind of thing, which contributed to the high-ish failure rates. There used actually be two linear algebra modules; one was MA2007 which was pretty much a rehash of 1st year and affectionately known as "baby linear algebra". I think that one is gone though and so would I be right in saying that physics people do the MA2055 one (aka "proper linear algebra"!) that maths and maths/physics students do?
Anyway, I think linear algebra is more important for a physicist but mathematical modelling would probably be more enjoyable.
|
About:
Basic Properties of Real Numbers: The Power Rules for Exponents
Metadata
Name:
Basic Properties of Real Numbers: The Power Rules for Exponents
ID:
m21897
Language:
English
(en)
Summary:
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.
Objectives of this module: understand the power rules for powers, products, and quotients.
|
Red prices are already discounted. If you are due a discount ACE will bill you at the red price, or your discount off the original price, whichever is lowest.
Saxon Math Grade 5/4 Complete Home School Kit [SX9781591413479]
$89.50$82.34
Saxon Math Grade 5/4 Complete Home School Kit The Saxon Math Middle Grades textbooks move students from primary grades to algebra. Each course contains a series of daily lessons covering all areas of general math. Each lesson presents a small portion of math content (called an increment) that builds on prior knowledge and understanding. After an increment is introduced, it becomes a part of the student's daily work for the rest of the year. This cumulative, continual practice ensures that students will retain what they have learned. The home school kit for Math 5/4, Math 6/5, Math 7/6, and Math 8/7 consist of a student textbook with 120 lessons and 12 investigations, a Tests and Worksheets Booklet (which includes tests and fact practice worksheets), and a Solutions Manual (which offers step-by-step solutions to all lessons, investigations, and test questions). Algebra 1 2 includes a textbook, Home School Packet (31 test forms in addition to answers for all textbook problems and test questions), and a Solutions Manual.
|
*—"Core" math modules (PRE, ALG,
GEO, TRI) are sold in packages. Buy any three for $395, or
all four for $495.
Geometry (GEO)
EducAide's Geometry Database has been several years in the making—and well
worth the wait! This collection of more than 5000 problems is simply unmatched
in its breadth and depth of coverage, and the huge number of diagrams (more
than 800) makes the database even more attractive.
As a core module, the Geometry database is intended for regular classroom
instruction. The wide range of topics and question-types makes it suitable for
use with any textbook or course of study, including newer, integrated math
courses. Although curriculums vary widely in scope and sequence, the database's
topical organization makes it possible to locate just the right questions for
tests, review worksheets, and daily lessons.
In the database, you will find extensive coverage of both synthetic geometry
(proofs and logic) and analytic geometry (measurement, coordinate systems, and
connections to algebra). The question types include: short-answer, definitions,
fill-in-the-blank, true-false, sometimes-always-never, logic exercises, and
traditional two-column and paragraph proofs. In addition, algebra skills are
integrated into perhaps 20% of the problems, so that students are continually
working with variables and solving linear and quadratic equations and systems.
|
Topology Point-set and Geometric
9780470096055
ISBN:
0470096055
Pub Date: 2007 Publisher: Wiley & Sons, Incorporated, John
Summary: The to develop more sophisticated intuition and enabling them to le...arn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors. Along with the standard point-set topology topics-connected and path-connected spaces, compact spaces, separation axioms, and metric spaces-Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students. Topology: Point-Set and Geometric features: A short introduction in each chapter designed to motivate the ideas and place them into an appropriate context Sections with exercise sets ranging in difficulty from easy to fairly challenging Exercises that are very creative in their approaches and work well in a classroom setting A supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs[
|
Precalculus: Mathematics for Calculus Book Description
This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, so that when students finish the course, they have a solid foundation in the principles of mathematical thinking. This comprehensive, evenly paced book provides complete coverage of the function concept and integrates substantial graphing calculator materials that help students develop insight into mathematical ideas. The authors' attention to detail and clarity, as in James Stewart's market-leading Calculus text, is what makes this text the market leader.
Popular Searches
The book Precalculus: Mathematics for Calculus by James Stewart, Lothar Redlin, Saleem Watson
(author) is published or distributed by Brooks Cole [0495109975, 9780495109976].
This particular edition was published on or around 2005-10-27 date.
Precalculus: Mathematics for Calculus has Paperback binding and this format has 397
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Math Refresher Workshop
Math Refresher Workshop
Are you ready for college-level math?
This course is designed for college-bound students who want or need to strengthen their skills and knowledge of college-level Algebra. This class is perfect for the student who has not participated in a math course recently.
This workshop will help you:
Illustrate knowledge by solving practice problems
Develop critical thinking skills
Provide review of the key concepts of Algebra including but not limited to:
Real Numbers
Graphing and lines
Linear Equations
Exponents and Polynomials
Functions
Factoring
Date: Please check back for more 2013 dates!
Time: 8:00 a.m. - 12:30 p.m.
Cost: $75
Location: Aultman College of Nursing and Health Sciences
(Located on the corner of Dartmouth and 9th St. SW)
Space is limited – to reserve your seat, Register Now!
Aultman College Community Education - ACCE
Office Hours:
Monday – Thursday: 8a-5pm
Friday: 8a-4pm
330.363.6776
We are conveniently located on the 2nd floor of the Aultman Education Center in the Aultman College main office.
Sign up for the ACCE e-news!
Sign up here to join our email list and receive monthly e-newsletters with new course announcements and early registration promotions!
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An Algebra Subsystem for Diagnosing Students' Input in a Physics Tutoring System To help a student in an introductory physics course do quantitative homework problems, an intelligent tutoring system must determine information of an algebraic nature. This paper describes a subsystem which resolves such questions for Andes2. The capabilities of the subsystem would be useful for any ITS which deals with problems involving complex systems of equations. This subsystem is capable of 1) solving the systems of equations at the level of introductory physics problems, 2) checking the Author(s): Shapiro Joel A.
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Information Systems This textbook will teach students how to exploit IS in a technology-rich environment. It will emphasize why, no matter what their major, information and communications technologies (ICT) are, and increasingly will be, a critical element in their personal success and the success of their organizations. PDF file. Author(s): No creator setMarriage and Family Relationships This is an introductory course on marriage and the family, intended to present a more balanced understanding than your own personal experience might give you. A second objective is to apply what you learn in class to your own life, and better understand what you personally want in your future family and relationship experiences. Author(s): No creator set
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Family Finance Upon completion of this course you should be able to: 1. Identify personal/family values and establish appropriate financial goals. 2. Develop financial plans that reflect your values and goals. 3. Begin implementation of your plans to meet short and long term financial goals. 4. Evaluate options for providing financial security throughout your life. 5. Recall and apply specific fact concerning various financial topics, tools, and servicesWoz Presents the Apple Historical Museum Steve Wozniak, the inventor of the personal computer, provides us with a tour of historical Apple II products. He shows how he scrounged to get the first parts to construct his computer. The first computers did not have cases because they were to be truly portable. Author(s): No creator set
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Ground-Level Ozone: Your Vehicle In this activity, students quantify and analyze their personal contributions of smog-forming compounds due to driving. The activity builds upon the previous lesson (Ground-Level Ozone).License information
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Obliquity: Why Our Goals Are Best Pursued Indirectly. Many goals are more likely to be achieved when pursued indirectly: the most profitable companies are not the most aggressive in chasing profits and the wealthiest are not the most materialistic. By understanding the principle of Obliquity we can make better decisions in our personal and professional lives. Author(s): No creator set Author(s): Hawkins Jan,Pea Roy D.
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11.401 Introduction to Housing, Community and Economic Development (MIT) This class explores how public policy and private markets affect housing, economic development, and the local economy. It provides an overview of techniques and specified programs, policies, and strategies that are (and have been) directed at neighborhood development. It gives students an opportunity to reflect on their personal sense of the housing and community development process. And it emphasizes the institutional context within which public and private actions are undertaken. Author(s): Thompson, J. Phillip
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Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative C
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CARY, NC (Aug. 08, 2012) – SAS Curriculum Pathways has launched a free Algebra 1 course that provides teachers and students with all the required content to address the Common Core State Standards for Algebra. Available online, the course engages students through real-world examples, images, animations, videos and targeted feedback. Teachers can integrate individual components or use the entire course as the foundation for their Algebra 1 curriculum.
"Success in Algebra 1 opens the door to STEM opportunities in high school and beyond, and can set students on the path to some of the most lucrative careers," said Scott McQuiggan, Director of SAS Curriculum Pathways. "This course gives teachers engaging content to support instruction, and will help them meet Common Core requirements."
SAS developed the Algebra 1 course in collaboration with the North Carolina Virtual Public School, the North Carolina Department of Public Instruction and the Triangle High Five Algebra Readiness Initiative, an organization that promotes the important role mathematics teachers play in preparing students for college and careers.
The course maps to publisher requirements recently established by the lead writers of the Common Core State Standards for Mathematics. More specifically, the course addresses the authors' concerns for greater emphasis on mathematical reasoning, rigor and balance. In addition, the course takes a balanced approach to three elements the writers see as central to course rigor: conceptual understanding, procedural skill, and opportunities to apply key concepts. It incorporates 21st-century themes like global awareness and financial literacy while weaving assessment opportunities throughout the content.
While Algebra 1 is the first full course developed, SAS Curriculum Pathways provides interactive resources in every core subject for grades six through 12 in traditional, virtual and home schools at no cost to all US educators. SAS Curriculum Pathways has registered more than 70,000 teachers and 18,000 schools in the US.
SAS Curriculum Pathways aligns to state and Common Core standards (a framework to prepare students for college and for work, and adopted by 45 states), and engages students with differentiated, quality content that targets higher-order thinking skills. It focuses on topics where doing, seeing and listening provide information and encourage insights in ways conventional methods cannot. SAS Curriculum Pathways features over 200 Interactive Tools, 200 Inquiries (guided investigations, organized around a focus question), 600 Web Lessons and 70 Audio Tutorials.
SAS IN EDUCATION
In addition to SAS Curriculum Pathways online resources, SAS analytics and business intelligence software is used at more than 3,000 educational institutions worldwide for teaching, research and administration. SAS has more than three decades of experience working witheducational institutions.® .
Many new homeschoolers are often driven to stick to rigid school hours. Admittedly, when our family began, that was exactly what we believed. It took some time, observation, and the sound advice of some seasoned homeschoolers that helped us see the light. I had to ask myself why I was so resistant to changing in the first place. The answer was clear. Institutionalized thinking.
Institutionalized thinking is the idea that something cannot be done because it has never done before within a given set of parameters (i.e. classroom, industry,etc.). Most of us that are products of the public school system, universities, corporate America, etc. are victims. The side effects can linger long after we have been exposed and indoctrinated. Here are just a few of the symptoms:
Following rules, black and white thinking (not flexible, unable to perceive the value in gray areas).
Making assumptions – about others, about the world, about ideas, about the expectations you feel weighing on you, about your own abilities.
Over-reliance onlogic, along with assuming you have an accurate grasp of what is logical.
I realized that just because" it" had always been that way, didn't mean that "it" had to continue to be that way. I began my quest to be more flexible by alternating our school hours. I introduced more field trips and unique ways to approach lessons. I began to embrace every teachable moment that I could.
So what is a teachable moment? A teachable moment is that moment when a unique, high interest situation arises that lends itself to discussion of a particular topic. For example, you are teaching a lesson about the seven continents and your child expresses a particular interest in the Panama Canal. You can embrace this teachable moment and delve deeper into the area of interest. You begin to talk about imports and exports and so on. Is it a tangent? Sort of, but your child is more likely to retain what he/she learns because of their interest in the subject~Steve Jobs, 2005
Teachable moments can occur at any moment, any place, anytime, so embrace them! They help restore the zeal for teaching your child and affirm you as a capable educator.
Blogroll
Disclaimer
All postings and emails are not intended to be legal advice and are distributed for information purposes only. Additionally, they are not intended to be and do not constitute the giving of legal advice.
|
MATH 112. Mathematics for Elementary Education II
Description:
This course is a continuation of Math 111, with the same philosophy and emphasis on achieving a deep understanding of elementary school mathematics. The content for Math 112 includes the real number system (as comprised of terminating, repeating, and non-repeating decimals), percents and proportions, probability, descriptive statistics, measurement (in English, metric, and non-standard units), and an overview of basic terminology and concepts from geometry. Intended for elementary education majors.
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Number systems
Number systems and the rules for combining numbers can be daunting. This...
Number
After studying this unit you should be able to:
understand the arithmetical properties of the rational and real numbers;
understand the definition of a complex number;
perform arithmetical operations with complex numbers;
represent complex numbers as points in the complex plane;
determine the polar form of a complex number;
use de Moivre's Theorem to find the nth roots of a complex number and to find some trigonometric identities;
understand the definition of ez, where z is a complex variable;
explain the terms modular addition and modular multiplication;
use Euclid's Algorithm to find multiplicative inverses in modular arithmetic, where these exist;
explain the meanings of a relation defined on a set, an equivalence relation and a partition of a set;
determine whether a given relation defined on a given set is an equivalence relation by checking the reflexive, symmetric and transitive properties;
understand that an equivalence relation partitions a set into equivalence classes;
Number systems
Introduction properties. You will meet other properties of these numbers in the analysis units, as the study of real functions depends on properties of the real numbers. We note that some quadratic equations with rational coefficients, such as x2 = 2, have solutions which are real but not rational.
In Section 2 we introduce the set of complex numbers. This system of numbers enables us to solve all polynomial equations, including those with no real solutions, such as x2 + 1 = 0. Just as real numbers correspond to points on the real line, so complex numbers correspond to points in a plane, known as the complex plane.
In Section 3 we look further at some properties of the integers, and introduce modular arithmetic. This will be useful in the group theory units, as some sets of numbers with the operation of modular addition or modular multiplication form groups.
In Section 4 we introduce the concept of a relation between elements of a set. This is a more general idea than that of a function, and leads us to a mathematical structure known as an equivalence relation. An equivalence relation on a set classifies elements of the set, separating them into disjoint subsets called equivalence classes.
This unit is an adapted extract from the Open University course
Pure mathematics
(M208) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]
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Strategic Math
Course Grade Level Description
Introduction to Strategic Math
This course is required for students whose mathematics proficiency is below grade level as indicated by eighth grade diagnostic assessments and teacher recommendation. This course meets daily for 90 minutes and is taken at the same time as their Integrated Algebra I class.
Strategic Math Goal Statement
This course is designed to help students improve the transition from arithmetic and pre-algebra understandings to algebraic understandings. This course is designed for students to increase the skills needed to be successful in Algebra. The balanced approach to mathematics in this course will provide diagnostic interventions based on individual student needs, and provide support for the Integrated Algebra I class.
Additional benefits of this course are:
targeted support to students so that they are successful in Algebra
boost self-confidence in mathematics
review/preview Algebra concepts
show students how effort is related to success in mathematics
strengthen vocabulary
Data from the Algebra course and collaboration amongst the Algebra/Strategic Mathematics teachers will be used to inform instruction.
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Many languages, many cultures, one goal—high-quality mathematics education…More]]>14118Thu, 19 Apr 2012 00:00:00 GMTDeveloping Essential Understanding of Mathematical Reasoning for Teaching Mathematics in Grades Pre-K-8 (List Price $32.95 Member Price $26.36)794Tue, 25 Oct 2011 00:00:00 GMTCommon Core Mathematics in a PLC at Work, Grades 3-5 3–5. Discover what students should learn and how they should learn it at each grade level, including deep support for the unique work for Number & Operations—Fractions in grades 3–5 and learning progression models that capstone expectations for middle school mathematics readiness.
By connecting the CCSSM to previous standards and practices, the book serves as a valuable guide for teachers and administrators in implementing the CCSSM to make mathematics education the best and most effective for all students.
More]]>14327Wed, 30 Jan 2013 00:00:00 GMTEyes on Math: A Visual Approach to Teaching Math Concepts (List Price $29.95 Member Price $23.96)
A unique teaching resource that provides engaging, full-color graphics and pictures with text showing teachers how to use each image to stimulate mathematical teaching conversations around key K–8 concepts.
Copublished with Teachers College PressMore]]>14573Mon, 17 Dec 2012 00:00:00 GMTDeveloping Essential Understanding of Statistics for Teaching Mathematics in Grades 6-8 (List Price $36.95 Member Price $29.56)
This book focuses on the essential knowledge for mathematics teachers about statistics. It is organized around four big ideas, supported by multiple smaller, interconnected ideas--essential understandings.More]]>13800Mon, 25 Feb 2013 00:00:00 GMTAdministrator's Guide: Interpreting the Common Core State Standards to Improve Mathematics Education (List Price $23.95 Member Price $19.16)More]]>14288Wed, 12 Oct 2011 00:00:00 GMTDeveloping Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5 (List Price $32.95 Member Price $26.36)
Help your upper elementary school student develop a robust understanding of rational numbers. More]]>13493Tue, 12 Oct 2010 00:00:00 GMTDeveloping Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3-5 (List Price $32.95 Member Price $26.36) More]]>13796Tue, 19 Apr 2011 00:00:00 GMTDeveloping Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3-5 (List Price $32.95 Member Price $26.36)
Move beyond the mathematics you expect your students to learn More]]>13795Tue, 19 Apr 2011 00:00:00 GMTDeveloping Essential Understanding of Geometry for Teaching Mathematics in Grades 9-12 (List Price $35.95 Member Price $28.76)
This book focuses on essential knowledge for teachers about geometry. It is organized around four big ideas, supported by multiple smaller, interconnected ideas--essential understandings.
More]]>14123Thu, 19 Apr 2012 00:00:00 GMTDeveloping Essential Understanding of Proof and Proving for Teaching Mathematics in Grades 9-12 (List Price $35.95 Member Price $28.76)
This book focuses on essential knowledge for teachers about proof and the process of proving. It is organized around five big ideas, supported by multiple smaller, interconnected ideas—essential understandings. More]]>13803Wed, 17 Oct 2012 00:00:00 GMTDefining Mathematics Education – Presidential Yearbook Selections 1926-2012 (List Price $59.95 Member Price $47.96)
The 75th Anniversary Yearbook: Celebrating a Valued Tradition of Defining Mathematics Education
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This book examines five big ideas and twenty-four related essential understandings for teaching statistics in grades 9–12. More]]>13804Thu, 14 Feb 2013 00:00:00 GMTRich and Engaging Mathematical Tasks: Grades 5-9 (List Price $36.95 Member Price $29.56)
A valuable resource to any mathematics teacher, this rich collection of mathematical tasks will enliven students' engagement in mathematical thinking and reasoning and help them succeed in the classroom. More]]>13516Wed, 07 Mar 2012 00:00:00 GMTFocus in Grade 2
More]]>13790Tue, 05 Apr 2011 00:00:00 GMTDeveloping Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K-Grade 2 (List Price $32.95 Member Price $26.36)
Move beyond the mathematics you expect your students to learn. More]]>13792Tue, 25 Jan 2011 00:00:00 GMTDeveloping Essential Understanding of Number and Numeration for Teaching Mathematics in Pre-K-2 (List Price $29.95 Member Price $23.96)
Move beyond the mathematics you expect your students to learn. More]]>13492Wed, 28 Apr 2010 00:00:00 GMTMore Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction (List Price $29.95 Member Price $23.96)
Differentiate math instruction with less difficulty and greater success! More]]>13782Thu, 15 Apr 2010 00:00:00 GMTFocus in High School Mathematics: Reasoning and Sense Making (List Price $36.95 Member Price $29.56)
A framework to guide the development of future 9–12 mathematics
curriculum and instruction. More]]>13494Tue, 06 Oct 2009 00:00:00 GMTMathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction, and Learning (List Price $38.95 Member Price $31.16)
Award-winning author Page Keeley and mathematics expert Cheryl Rose Tobey apply the successful format of Keeley's best-selling Science Formative Assessment to mathematics. They provide 75 formative assessment strategies and show teachers how to use them to inform instructional planning and better meet the needs of all students. Research shows that formative assessment has the power to significantly improve learning, and its many benefits include:
Copublished with Corwin
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What tasks can you offer—what questions can you ask—to determine what your students know or don't know—and move them forward in their thinking?
This book focuses on the specialized pedagogical content knowledge that you need to teach fractions effectively in grades 3–5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with fractions—not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts.
More]]>14542Fri, 12 Apr 2013 00:00:00 GMTTeaching Mathematics through Problem Solving: Prekindergarten–Grade 6 (List Price $8.76 Member Price $8.76)
This volume and its companion for grades 6–12 furnish the coherence and
direction that teachers need to use problem solving to teach
mathematics.
More]]>12576Tue, 04 Nov 2003 00:00:00 GMTMath Jokes 4 Mathy Folks (List Price $11.95 Member Price $9.56)
Intended for all math types. Provides a comprehensive collection of math humor, containing over 400 jokes. It's a book that all teachers from elementary school through college should have in their library. More]]>13837Tue, 27 Apr 2010 00:00:00 GMTNavigating through Number and Operations in Grades 3-5 (with CD-ROM) (List Price $46.95 Member Price $37.56)
Activities in this book invite students to use fraction circles to compare fractions and dot arrays to explore multiplication and the distributive property. The authors present many other hands-on approaches as well. More]]>12952Wed, 28 Mar 2007 00:00:00 GMTAdministrator's Guide: How to Support and Improve Mathematics Education in Your School (List Price $20.95 Member Price $16.76)
Describes what administrators need to know about mathematics education
and how to support and improve mathematics education in their
schools.
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Copublished with Solution Tree PressMore]]>14383Wed, 09 May 2012 00:00:00 GMTNavigating through Geometry in Grades 3–5 (with CD-ROM) (List Price $37.95 Member Price $30.36)
The "big ideas" of geometry–shape, location, transformations, and spatial visualization–are the focus of this book. More]]>12173Sat, 01 Sep 2001 00:00:00 GMTNavigating through Measurement in Grades 3–5 (with CD-ROM) (List Price $41.95 Member Price $33.56)
This book follows students' natural progression from measuring with informal or nonstandard units to using standard units to measure such attributes as length, weight, angle, and temperature. More]]>12525Wed, 02 Feb 2005 00:00:00 GMTThe Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices (List Price $36.95 Member Price $29.56)
Each teacher and student brings many identities to the classroom. What is their impact on the student's learning and the teacher's teaching of mathematics?
This book invites K–8 teachers to reflect on their own and their students' multiple identities. Rich possibilities for learning result when teachers draw on these identities to offer high-quality, equity-based teaching to all students. Reflecting on identity and re-envisioning learning and teaching through this lens especially benefits students who have been marginalized by race, class, ethnicity, or gender. More]]>14119Wed, 10 Apr 2013 00:00:00 GMTAchieving Fluency: Special Education and Mathematics (List Price $36.95 Member Price $29.56)
"Is it a learning disability or a teaching disability?"
Achieving Fluency presents the understandings that all teachers need to play a role in the education of students who struggle: those with disabilities and those who simply lack essential foundational knowledge. This book serves teachers and supervisors by sharing increasingly intensive instructional interventions for struggling students on essential topics aligned with NCTM's Curriculum Focal Points, the new Common Core State Standards for Mathematics, and the practices and processes that overlap the content. These approaches are useful for both overcoming ineffective approaches and implementing preventive approaches.
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|
Connections in Mathematics
Course Description:
This course provides students with introductory experiences in symbolic logic, binary and other bases, probability, conditional probability, set theory, non-routine problem solving, topics of personal finance and investment, and the calculus necessary to participate in the Senior Science Scenario project during the final six weeks of the year. Both EXCEL and a variety of website-based applications will be used throughout the year. Emphasis is placed on conceptual understanding, solving real world applications, and fostering mathematical reasoning and communication.
Course Materials:
The text for this course will be "The Nature of Mathematics", 10th ed., by Smith (no relation). You will be responsible for the text while it is in your possession.
Methodology:
The beginning of most class periods will be used to answer questions on the material that is due for that day. The rest of the class period will consist of a variety of activities which will include lecture, individual and group problem solving, and exploration of questions and concepts. It is strongly advised that you prepare for each class meeting by working assigned homework problems and by reading and taking notes on the text to be covered in the next class meeting.
Study Aids:
There are many reference books and web sites widely available that can serve as study aids for this course. However, it is unlikely that any materials beyond those provided in class will be necessary. If you feel at any time that you require additional assistance, please discuss this with me at the beginning or end of the next class meeting.
Participation:
You should plan to be actively involved in class. This means being attentive and participating in class discussions and activities.
Absences (consult the Student Handbook for additional information):
When you miss any amount of class time, for any reason, it is your responsibility to contact a student colleague in the class to obtain the information you missed.
Foreseeable absences for any reason need to be discussed with the instructor in advance. Failure to do so will result in an unexcused absence.
If a student is absent (excused) for only one class meeting, upon return he/she is expected to have completed the work which was due on the day of absence. If a test was missed, the student is expected to take the test on the day of return. If a student misses two or more consecutive class meetings, he/she should talk to the instructor to devise a plan to catch up.
Work missed because of an unexcused absence cannot be made up. If a test is missed because of an unexcused absence, then that test grade will be lowered by 10 points for each day late.
Tardiness (consult the Student Handbook for additional information):
You are expected to be in our class, ready to learn, by our starting time. Given my responsibilities as the Director of the Governor's School, I might not be in the room; that does not relieve you of your responsibility to be in the class, ready to learn, by the beginning of class. I will permit one unexcused tardy without any grade penalty. After that, I will lower your semester grade by ½ a point for each unexcused tardy.
Honor Code:
Students are required to pledge all work that they turn in for a grade. Refer to CVGS Student Handbook for complete requirements.
Grading:
The grading scale is a standard 11/10/10/10 point scale.
Percentage and Grade Equivalent:
89.5-100
A
79.5-89.4
B
69.5-79.4
C
59.5-69.4
D
Earning less than 60 points will result in a failing grade for the course.
Course Description (First Semester):
During the first semester students will work with introductory experiences in symbolic logic, binary and other bases, voting methods, apportionment schemes and paradoxes, probability, conditional probability, set theory, and non-routine problem solving. Emphasis is placed on conceptual understanding, solving real world applications, and fostering mathematical reasoning and communication.
Specific Course Content and Objectives (First Semester):
the student will be able to:
translate sentences to symbolic form,
construct truth tables,
state the converse, inverse and contrapositive of statements,
determine the validity of an argument,
design a simple circuit (or a gate) as a logic application,
understand and use basic set theory concepts, including intersections, unions, complements, distributive and De Morgan's laws, and cardinality,
recall and be able to compare and contrast voting methods, voting dilemmas, apportionment methods and paradoxes
convert numbers in the decimal, binary, octal, and hexadecimal systems,
Class Participation (30 pts): Asking or answering questions well, putting problems on the board, and generally being attentive and engaged
It is your responsibility to keep track of the points you have earned and the assignments you have completed. Progress reports will report progress for the entire semester thus far. To reiterate, all grades will be cumulative from the beginning of the semester!
Tentative Course Schedule for First Semester:
We will follow the sequence of topics below, although adjustments will be made depending on how quickly we are able to move as a group.
2.1: Symbolic Logic
2.2: Truth Tables and Conditionals
2.3: Operators and Laws of Logic
2.4: Logical Proof
2.6: Logic Circuits
16.1: Voting Methods
16.2 Voting Dilemmas
16.3: Apportionment
16.4: Apportionment Paradox
Project #1
3.4: Binary + Octal + hexadecimal
10.1: Sets, subsets, Venn diagrams
10.2: Sets—combined operations, DeMorgan's Laws
10.3: Permutations
10.4: Combinations
10.5: Complex Counting
Project #2
11.1: Probability
11.2: Math Expectation
11.3 Probability Models
11.4: Calculated Probabilities
Project #3
15.1: Euler Circuits and Hamiltonian Cycles
15.2: Trees and Minimum Spanning Trees
Special Topics
REVIEW FOR EXAMEXAMS
Course Description (Second Semester)
This course provides students with experiences in topics of personal finance and investment and the use of EXCEL to facilitate calculations such as those used in an amortization schedule. Students will also learn the calculus basics necessary to participate in the Senior Science Scenario project during the sixth six weeks. The use of EXCEL and the free website will be explored.
Throughout the course emphasis is placed on conceptual understanding, solving real world applications, and fostering mathematical reasoning and communication.
Specific Course Content and Objectives (Second Semester): the student will be able to:
Class Participation: Asking or answering questions well, putting problems on the board, and generally being attentive and engaged is expected. Failing to do so can result in a deduction of points from the total points earned.
S-Cubed grade (1@150 pts)
|
Authors
Abstract
Problem-solving processes should extend beyond mere working problems by type where students are provided
algorithmic approaches to fit situations (e.g., rate, mixture, coin, investment, work) since reducing these typical problem
situations to "algorithmic processing" is counter-productive relative to higher-level problem-solving goals (NCTM, 1989,1991,
1996). By incorporating technological tools (CABRI Geometry II, spreadsheets, and graphing calculators) coupled with the
problem solving principles espoused by George Polya (famous mathematician and teacher of mathematics and mathematics
teachers), secondary school algebra problems can be taught as recommended by the NCTM curriculum standards to
appropriately meet recommended problem-solving goals. Even typical problems can therefore be used to expose students to
multiple problem-solving approaches that extend understanding and meta-cognitive abilities.
|
Synopsis
Translated into many languages, this book has been the standard university-level text for decades. Revised and enlarged by the author in 1952, it offers today's students exercises in construction problems, similitude, and homothecy, properties of the triangle and the quadrilateral, harmonic division, and circle and triangle geometry
|
Pages
Wednesday, May 1, 2013
This
column continues my report on results of the MAA National Study of
Calculus I, Characteristics
of Successful Programs in College Calculus.
This month I am
sharing what we learned about the use of graphing calculators (with
or without computer algebra systems) and computer software such as
Maple
or Mathematica.
Our results draw on three of the surveys:
Student
survey at start of term: We asked students how calculators and/or
computer algebra systems (CAS) were used in their last high school
mathematics class and how comfortable they are in using these
technologies.
Student
survey at end of term: We asked students how calculators or CAS had
been used both in class and for out of class assignments.
Instructor
survey at start of term: We asked instructors what technologies
would be allowed on examinations and which would be required on
examinations.
Our
first question asked students how calculators were used on exams in
their last high school mathematics class (see Figure 1). As in
previous columns, "research" refers to the responses of students
taking Calculus I at research universities (highest degree in
mathematics is doctorate), "undergrad" refers to undergraduate
colleges (highest degree is bachelor's), "masters" to masters
universities (highest degree is masters), and "two-year" to
two-year colleges (highest degree is associate's).
There
are several interesting observations to be made from this graph.
First, not surprisingly, almost all Calculus I students reported
having used graphing calculators on their exams at least some of the
time ("always" and "sometimes" were mutually exclusive
options). Second, there is a difference by type of institution.
Students at undergraduate colleges were most likely to have used
graphing calculators on high school exams (94%), then those at
research universities (91%), then masters universities (86%), and
finally two-year colleges (77%). The differences are small but
statistically significant. My best guess is that these are
reflections of the economic background of these students. A second
observation is that for most students, access to a graphing
calculator was not always allowed. However, it is still common
practice in high schools (roughly one-third of all students) to
always allow students to use graphing calculators on mathematics
exams.
Another
striking observation from Figure 1 is that the percentage of students
who were always allowed to use graphing calculators on exams is
almost identical to the percentage of students who were always
allowed to use graphing calculators with CAS capabilities on exams.
For all categories of students, over half of them were allowed to use
graphing calculators with CAS capabilities at least some of the time,
which suggests that over half of the students in college Calculus I
own or have had access to such calculators.
The
next graph (Figure 2) shows how students at the start of the term
reported their comfort level with using graphing calculators or
computer algebra systems (Maple
and Mathematica
were provided as examples of what we meant). The most interesting
feature of this graph is that students at two-year colleges are much
more likely to be comfortable with Maple
or Mathematica
than those at four-year programs. I suspect that the reason behind
this is that most Calculus I students at two-year colleges are
sophomores who took pre-calculus at that college the year before.
This gave them more opportunity to experience these computer algebra
systems.
Figure
2. Student attitude toward use of graphing calculator or CAS on a
computer such as Maple
or Mathematica.
The
graphs in Figures 3–5 show what students reported at the end of the
term about use of technology. For the graph in Figure 3, students
were asked how frequently each of these occurred in class. Percentage
shows the fraction of students who responded "about half the class
sessions," "most class sessions," or "every class session."
We note large differences in instructor use of technology generally
(for this question, "technology" was not defined), and especially
sharp differences for instructor use of graphing calculators or CAS
(with Maple and
Mathematica given as
examples). It is interesting that students are most likely to
encounter computer algebra systems in undergraduate and two-year
colleges, much less likely in masters and research universities.
Figure
3. End of term student reports on frequency of use of technology (at
least once/month). For this question, CAS refers to a computer
algebra system on a computer, such as Maple
or Mathematica.
The
first two sets of bars in Figure 4 show student responses to "Does
your calculator find the symbolic derivative of a function?" The
first set gives the percentage responding "N/A, I do not use a
calculator." The second set displays the percentage responding
"yes." Looking at the complement of these two responses, we see
that across all types of institutions, roughly 50% of students taking
Calculus I own a graphing calculator without CAS capabilities. The
third set records the percentage responding "yes" to the
question, "Were you allowed to use a graphing calculator during
your exams?" Note that there are some discrepancies between what
students and instructors report about allowing graphing calculators
on exams (Figures 4 and 6), but the basic pattern that graphing
calculators are allowed far less frequently at research universities
than at other types of institutions is consistently demonstrated.
Figure
4. End of term student reports on calculator use. No calculator = do
not use a calculator. Calculator with CAS = use a calculator with CAS
capabilities. Calc allowed on exams = graphing calculators were
allowed on exams.
We
also asked how often "The assignments completed outside of class
time required that I use technology to understand ideas." Again, we
see much less use of technology at research universities, the
greatest use at undergraduate and two-year colleges.
Figure
5. Frequency with which technology (either graphing calculators or
computers) was used for out of class assignments. Almost never = less
than once per month (includes never). Sometimes = at least once per
month but less than once per week. Often = at least once per week.
The
last two graphs (Figures 6 and 7) are taken from the instructor
responses at the start of the term: what technology they would allow
on their exams and what technology they would require on their exams.
Again, we see a clear indication that technology, especially the use
of graphing calculators without CAS capabilities, is much less common
at research universities than other types of institutions.
It
is interesting to observe that there are large numbers of instructors
who allow but do not require technology on the exams. At research
universities, 26% require the use of some kind of technology, and a
further 25% allow but do not require the use of some sort of
technology. For undergraduate colleges, 38% of instructors require
technology, an additional 42% allow it. At masters universities, 42%
require, and a further 33% allow. At two-year colleges, 52% require,
and an additional 36% allow.
Figure
6. Start of term report by instructor of intended use of technology
on exams. GC = graphing calculator. Most of those who checked "other"
reported that they allowed graphing calculators on some but not all
parts of the exam. Some reported allowing only scientific
calculators.
Figure
7. Start of term report by instructor of intended use of technology
on exams. GC = graphing calculator. Most of those who checked "other"
reported that they required graphing calculators on some but not all
parts of the exam. Some reported requiring only scientific
calculators.
We
see a pattern of very heavy use of graphing calculators in high
schools, driven, no doubt, by the fact that students are expected to
use them for certain sections of the Advanced Placement Calculus
exams. They are still the dominant technology at colleges and
universities, but there the use is as likely to be voluntary as
required. This implies that in many colleges and universities
questions and assignments are posed in such a way that graphing
calculators confer little or no advantage. The use of graphing
calculators at the post-secondary level varies tremendously by type
of institution. Yet even at the research universities, over half the
instructors allow the use of graphing calculators for at least some
portions of their exams.
Monday, April 1, 2013
Last month (MAA Calculus Study: Good Teaching) I discussed the student-described attributes
of instructors that were highly correlated with improvements in student
confidence, enjoyment of mathematics, and desire to continue to study
mathematics. This month I will discuss a second set of instructor attributes
that we are labeling "Progressive Teaching" because they are generally
associated with approaches to teaching and learning that focus on active
engagement of the students.
Here the evidence for improved results is less clear. In
particular, Sadler and Sonnert discovered a strong interaction with the
attributes we are calling "Good Teaching": teachers who rated high on Good
Teaching improved student outcomes if they also rated high on Progressive
Teaching. But if they rated low on Good Teaching, then a high rating on
Progressive Teaching had a strongly negative effect on student confidence. This
might have been expected. Good Teaching
describes student-teacher interactions, including the degree to which students
feel encouraged to participate in class and supported by the instructor. It is not surprising that students who are
encountering unfamiliar approaches to classroom learning react negatively if
they believe that that the instructor is not encouraging or supportive.
We also have evidence of some consistently positive effects
from Progressive Teaching. Even with a low score on Good Teaching, Progressive
Teaching was seen to be helpful in convincing students to continue the study of
mathematics. Our conclusions are that:
b.Good Teaching is more important to student persistence
than Progressive Teaching,
c.both can serve to improve student outcomes, and
d.teaching is most effective when instructors rate
high on both measures.
There were 12 student responses that clustered into what
we are calling Progressive Teaching:
My calculus instructor frequently 1.Assigned sections of the textbook to read before
coming to class.
2.Had students work with one another. 3.Had students give presentations. 4.Asked students to explain their thinking in
class. 5.Required students to explain their thinking on
homework assignments. 6.Required students to explain their thinking on
exams. 7.Held whole class discussions.
My calculus instructor did not frequently 8.Lecture.
Assignments completed outside of class 9.Required that I solve word problems. 10.Were
problems unlike those done in class or in the book. 11.Were
often submitted as a group project. 12.Were
returned with helpful feedback and comments.
With one exception, the following graphs show the percentage
of students who reported that their instructors employed each of these
practices often or very often (a 5 or 6 on a Likert scale from 1 = not at all
to 6 = very often). The exception is practice #8. Here we record the percentage
of students who responded 1, 2, or 3 on the same scale to the question, "During
class time, how frequently did your instructor lecture?".
We see that for most of the instructor behaviors (practices
1 through 8), the undergraduate colleges and two-year colleges are where these
are most likely to be employed. The relatively large percentage of instructors
at masters universities who had students give presentations in class (13% as
opposed to 6% at all other types of institutions) is still small and may be an
artifact of the relatively small number of responses from students at masters
universities (305 students at 18 institutions).
The research universities are where we find the most challenging
problems being posed on assignments, either word problems or those unlike those
done in class or in the book. Instructors at two-year colleges provide the most
helpful feedback on assignments, instructors at research universities the least
helpful feedback.
Friday, March 1, 2013
One of the primary goals of the MAA Calculus Study, Characteristics of Successful Programs in College Calculus (NSF #0910240), has been to
identify the factors that are highly correlated with an improvement in student
attitudes from the start to the end of the calculus course: confidence in
mathematical ability, enjoyment of mathematics, and desire to continue the
study of mathematics. To this end, Phil Sadler and Gerhard Sonnert of the
Science Education Department within the Harvard-Smithsonian Center for
Astrophysics constructed a hierarchical linear model from our survey responses
to identify these factors. The factors reside at three levels: institutional,
classroom, and individual student. Not surprisingly, most of the variation in
student attitudes can be explained by student background, but there are
influences at the institutional and classroom level. We have been particularly
interested in what happens at the classroom level where there is the greatest
opportunity for improvement.
Sadler and Sonnert ran a factor analysis of the
classroom-level variables, clumping those responses that were highly correlated.
They discovered that the responses broke into three distinct clusters, which we
are labeling "technology," "progressive teaching," and "good teaching" because
these seem to describe the characteristics of the instruction. By far, the most
important of these in terms of high correlation with improved attitudes is
"good teaching." Listed below are the 21 student-reported characteristics of
instruction that are highly correlated with each other and highly correlated
with improvements in student attitudes, characteristics that collectively we
are calling "good teaching":
My calculus instructor:
Asked questions to determine if I understood what was being
discussed.
Listened carefully to my questions and comments.
Discussed applications of calculus.
Allowed time for me to understand difficult ideas.
Helped me become a better problem solver.
Encouraged students to enroll in Calculus II.
Acted as if I was capable of understanding the key ideas of
calculus.
Made me feel comfortable asking questions during class.
Encouraged students to seek help during office hours.
Presented more than one method for solving problems.
Made class interesting.
Provided explanations that were understandable.
Was available to make appointments outside of
office hours, if needed.
My calculus instructor did not:
Discourage me from wanting to continue taking calculus.
Make students feel nervous during class.
My instructor often or very often:
Showed how to work specific problems.
Asked questions.
Prepared extra material to help students understand calculus
concepts or procedures.
In addition:
My calculus exams were a good assessment of what I learned.
My exams were fairly graded.
My homework was fairly graded.
The good news is that most calculus instructors rated highly
on most of these characteristics. This good news needs to be tempered by two
facts: Instructors could and in many cases did elect not to participate even
though other instructors at their institution were involved in the study, and these
responses were all collected at the end of the term. They reflect the opinions
of the students who had successfully navigated this course, predominantly
students who were earning an A or a B in the course (roughly 40% A, 40% B, 20%
C).
It is interesting and informative to see how students at
different types of institutions rated their instructors on these criteria. We
followed CBMS in categorizing post-secondary institutions by the highest
mathematics degree offered at that institution. I am using "research" to
designate universities that offer a PhD in Mathematics (predominantly large
state flagship universities), "masters" if the highest degree is a master's
(predominantly public comprehensive universities), "undergrad" if it is a
bachelor's degree (predominantly private liberal arts colleges), and "two-year"
if it is an associate's degree (predominantly community and technical colleges).
As shown in the graphs at the end of this article, instructors at research
universities got the lowest ratings on every characteristic except "showed how
to work specific problems." For most of these characteristics, instructors at undergraduate
colleges were the next lowest, then masters universities, and most of the time
instructors at two-year colleges received the highest ratings.
There were a few notable exceptions. Instructors at undergraduate
colleges received the highest ratings in some of the areas where one would
expect them to be strong:
Acted as if I was capable of understanding the key ideas of
calculus.
Encouraged students to seek help during office hours.
Was available to make appointments outside of office hours,
if needed.
Did not make students feel nervous during class.
Masters universities scored highest in often or very often
showing how to work specific problems, and just barely edged out two-year
colleges in "listened carefully" and "my exams were fairly graded."
There are a number of possible explanations for the
weaknesses of research universities and the strengths of two-year colleges. One
is class size. The largest classes are found at the research universities where
average class size is 53, the smallest at two-year colleges where the average
is 21. However, average class size at masters universities is larger than at
undergraduate colleges, so class size cannot be the only explanatory variable. Some
of the discrepancies between institution types may be explained by student
expectations. This is because SAT scores and high school mathematics GPA are
highest for research universities, then undergraduate colleges, then masters
universities, and lowest for two-year colleges. Better students may have higher
expectations of their instructors, or they may be more discouraged by
encountering difficulties in this course. The differences may also have
something to do with age and thus maturity of the students. The youngest
students are at research universities, the oldest at two-year colleges. They
also may be related to the relatively large number of instructors at research
universities who teach calculus but have little or no interest in teaching this
course, as opposed to two-year colleges where the interest is very high (see my
November column, MAA Calculus Study: The Instructors). Nevertheless, it is discouraging that
students at research universities seem to be getting calculus instruction that has
a worse effect on student attitudes than instruction at other types of
institutions.
Friday, February 1, 2013
The National Research Council of the National Academies has
just released the preliminary version of its report, The Mathematical Sciences in 2025[1]. This was produced in response to a request from the National
Science Foundation. It comes as the latest in a series of glimpses into the
future of mathematics that go back to the "David reports" of 1984 and 1990 [2,3]
and the "Odom study" of 1998 [4]. This report is important because it will
influence the direction NSF takes as it plans for the future.
The emphasis of the report is on the central role that the
mathematical sciences are taking within research in areas as diverse as
biology, finance, and climate science. Traditional disciplinary boundaries are
blurring. There is an increasing need for scientists who are well grounded in
mathematical sciences, especially the statistical and computational sciences,
as well as other disciplines. This goes two ways. It means opening courses and
programs in the mathematical sciences, especially at the graduate level, to
those in other fields of study, and it means ensuring that students graduating in
the mathematical sciences are prepared to work in this interdisciplinary world.
This has implications right down the line of mathematics
education. The authors of the report question whether, in a scientific world
that is dominated by big data and the challenges of large-scale computation,
the traditional calculus-focused curriculum is the most appropriate for all
students. As they say, "Different pathways are needed for students who may go
on to work in bioinformatics, ecology, medicine, computing, and so on. It is
not enough to rearrange existing courses to create alternative curricula; a redesigned offering of courses and majors
is needed [my emphasis]." (NRC 2013, p. S-9)
The report also stresses the importance of attracting more
women and students from traditionally underrepresented minorities to the mathematical
societies. This is the one place where I disagree with the report, for it
asserts that, "While there has been progress in the last 10–20 years, the
fraction of women and minorities in the mathematical sciences drops with each
step up the career ladder." (NRC 2013, p. S-10). I don't question the drop. I
question whether there has been progress over the last 10–20 years.
If we look at mathematics majors (bachelor's degrees) by
gender, we see that over the period 1990 to 2011 the number of men majoring in
mathematics grew by 25% while the number of women grew by only 10% (Figure 1).
As a result, the percentage of bachelor's degrees in the mathematical sciences
going to women has dropped to 43.1%, the first time it has been this low since
1981. This is having knock-on effects for graduate programs. The percentage of
bachelor's degrees in mathematics that went to women peaked in 1999 at 47.8%.
The percentage of master's degrees in mathematics that went to women peaked in
2004 at 45.1% and has since dropped back to 40.9%. The percentage of doctoral
degrees in mathematics that went to women peaked in 2008 and '09 at 31.0%. It
has since dropped back to 28.6%. The good news is that the past decade has seen
strong growth in the number of mathematics majors, but two-thirds of the growth
since 2001 has been in the number of men.
We see an even more discouraging pattern among Black students
(Figure 2). The number of Black mathematics majors is essentially back to where
it was twenty years ago despite the number of bachelor's degrees earned by
Black students almost tripling over this period. The number of Black
mathematics majors peaked in 1997 at 1,089. It was back down to only 840 in
2011. The number of ethnically Asian mathematics majors has been growing
strongly over the past decade. Even so, the number earning undergraduate degrees
in the mathematical sciences has only doubled since 1990, while the number
earning bachelor's degrees has tripled. The growth in the number of Hispanic
mathematics majors looks good, having slightly more than tripled in twenty
years, until you realize that the number of Hispanic students graduating from
college is almost five times what it was in 1990 (154,000 versus 33,000). Where
we do see strong growth, especially since 2007, is in the number of
non-resident aliens majoring in mathematics, which now stands at 7% of all US
mathematics majors.
I must emphasize that the NRC report does highlight the
importance of increasing the participation of women and members of
underrepresented groups. It includes the following specific recommendation:
Recommendation
5-4:
Every academic department in the mathematical sciences should explicitly
incorporate recruitment and retention of women and underrepresented groups
into the responsibilities of the faculty members in charge of
the undergraduate program, graduate program, and faculty hiring and
promotion. Resources need to be provided to enable departments to adopt,
monitor and adapt successful recruiting and mentoring programs that have
been pioneered at other schools and to find and correct any disincentives
that may exist in the department. (NRC 2013, p. 5-18)
I have only touched on a few of the topics covered in the NRC
report. It also discusses the increasingly important role of the mathematical
sciences institutes, the issue of maintaining online repositories of mathematical
research such as arXive, and the threats to
mathematics departments as more instruction—especially for the service courses
that often provide the justification for a large mathematics faculty—is moved
online. This is a report well worth reading and pondering.
Tuesday, January 1, 2013
Two things happened in the week before Christmas that got me
thinking about grade inflation. The first was that I graded the final exams for
my multivariable calculus class. I have never before seen my students do so
well. Out of 33 students in the class, 22 received an A. For my class, an A
requires earning more than 92% of the total possible grade. The last time I
graded on a curve was over 20 years ago.
This past semester I had worked these students hard. They
were responsible for and graded on:
Reading Reflections (three times per week, reading the
section and answering questions about the material before we discussed it in
class).
Two sets of homework each week (about 12 fairly
straightforward questions on WeBWorK due on Thursdays and three challenging
multi-part problems due on Mondays).
Seven short projects developed by Tevian Dray and Corinne
Manogue as part of their Bridge Project (see
These were started in groups of three or four, but each student was responsible
for writing his or her own three to five page report of the solution. For the
first report, I required a first draft that was critiqued and returned for
revision and resubmission.
A major project based on the Hydro-Turbine Optimization
chapter in Applications of Calculus
[1]. The project was started in groups. Each student was responsible for an
8–12 page paper explaining the solution. The papers were turned in, critiqued,
and returned for revision and resubmission. LaTeX and pdf files of my version
of this project are available here.
Two examinations during the semester and a final exam. After
each exam during the semester, students were required to write about the
problems they had missed points on, explain what they did wrong, and explain
how to do it correctly. They could earn back half the points they had lost. For
the final exam, they had to explain what they were doing to solve the problems,
not just give an answer.
I was available to my students every afternoon, and I also had
a great undergraduate preceptor (teaching assistant) who held help sessions
Sunday and Thursday evenings, before the homework assignments were due. By the
end of the semester, over half the class was coming to each of these, and so
she organized them into groups working with each other on the homework while
she circulated to help the groups that were stuck.
Not surprisingly, in the end of semester course evaluations my
students wrote about how much work they had done for this course. And yet, when
asked specifically whether or not they agreed with the statement, "The general
workload was appropriate for this level course," only five of my 33 students
disagreed. One student comment that summarized the tenor of the end of course
evaluations stated, "I would say that the course is difficult and a lot of
work, but very rewarding, because if you put in a lot of time and effort then
you can see yourself understand the material and do well. Although the course
can be really hard at times, there is always somewhere to go for help."
The second thing that happened this past week was my
discovery of How Learning Works: 7
Research-Based Principles for Smart Teaching [2]. This collaborative
effort, published in 2010, translates what has been learned by those engaged in
research in undergraduate education into practical guidance for those of us in
the classroom. What the authors call principles, I see more as facets of
teaching to which I need to pay attention. This is my own paraphrasing of these
principles or facets:
The need to understand the variety
of prior knowledge that my students bring to my class and how it helps or
hinders them.
The importance of how students
organize the knowledge they are acquiring and the need for me to understand
common misalignments and to help them make the necessary connections.
The critical role of student
motivation and my responsibility to strengthen it.
The need to develop automaticity
in basic skills and the fact that learning how to integrate and apply these
skills requires guidance and directed practice from me.
How important it is that I provide
useful feedback that is targeted at improving performance.
The role of the social, emotional,
and intellectual climate in my classroom.
The need for me to guide students in
practicing metacognition, monitoring what they are doing and why.
The book discusses the relevant research, but is also full
of examples of traps we can fall into and strategies for dealing with these
principles or facets in order to improve our teaching.
One trap discussed under #3 describes the teacher who, with
the intent of spurring his students to work hard, warned them at the start of
the course that they could expect that a third of them would not pass. This had
exactly the opposite effect. With the expectation that they would not do well
regardless of how much effort they put into the course, a large proportion of
the students directed their time and energy to other courses.
The issue here is motivation, getting the students to put in
the effort needed to learn the material. I believe that I did succeed
particularly well this past semester in motivating most of my multivariable
calculus students. How Learning Works identifies three levers that motivate students
to work hard. The first is value.
They have to believe that what I want them to learn will be of value to them.
Personal enthusiasm on my part goes a long way toward building this sense of
value. The second is a supportive
environment. They have to believe that the course is structured in such a
way as to help them be successful, rather than throwing up obstacles to their
success. Starting the projects and encouraging them to share their
understanding of homework problems within groups, providing feedback and
multiple opportunities to demonstrate understanding (as with WeBWorK and the chance
to earn back points lost on exams), and the availability of myself and my preceptor
build the sense of support. The third is self-efficacy,
belief that one is capable of achieving success.
This last is the main reason I will never again grade on a
curve. The message sent by grading on a curve is that the proportion of
failures has been determined in advance, regardless of how much work students
are prepared to invest in the course. It is also why I am disturbed that in our
national survey of calculus, faculty at the start of the term were able to
predict, almost perfectly, what their grade distributions would be at the end
of the term (see the last bullet under Instructor Attitudes in The Calculus I
Instructor, Launchings, June 2011). Going into this course, I would never
have predicted 67% A's. I am delighted that what I did worked so well with so
many of my students. [3]
Which brings me back to the issue of grade inflation. Grade
inflation is a red herring because it misdirects our attention from what should
be our true concerns: What do our grades mean in terms of expectation of
student achievement and understanding? And how can we support as many students
as possible to meet our highest expectations?
[1] Straffin, P. D., Jr. 1996. Hydro-Turbine Optimization. Pages 240–250 in Applications of Calculus. P.D. Straffin, Jr., editor. Classroom Resource Materials. MAA. [2] Ambrose, S. A., M. W. Bridges,
M. DiPietro, M. C. Lovett, M. K. Norman. 2010. How Learning Works: 7 Research-Based Principles for Smart Teaching.
Jossey-Bass. [3] Not all my students did well. The class GPA was 3.5. What was
important was that I had explicit expectations of what would constitute A work,
that I clearly communicated what was required to meet those expectations, that
students saw them as challenging but achievable, and that my students really
were graded according to these expectations.
Saturday, December 1, 2012
This past spring, the National Research Council of the
National Academies released its report, Discipline-Based Education Research: Understanding and Improving
Learning in Undergraduate Science and Engineering[1]. The charge to the
committee writing this report was to synthesize existing research on teaching
and learning in the sciences, to report on the effect of this research, and to identify
future directions for this research. The project has its roots in two 2008
workshops on promising practices in undergraduate science, technology,
engineering, and mathematics education.
Unfortunately, between 2008 and 2012 undergraduate mathematics
education dropped out of the picture. The resulting report discusses
undergraduate education research only for physics, chemistry, engineering,
biology, the geosciences, and astronomy. Nevertheless, it is an interesting
report with useful information—especially the instructional strategies that
have been shown to be effective—that is relevant for those of us who teach
undergraduate mathematics.
The studies that are described are founded on the assumption
that students must build their own understanding of the discipline by applying
its methods and principles, and this is best accomplished within a
student-centered approach that puts less emphasis on simple transmission of
factual information and more on student engagement with conceptual
understanding, including active learning in the classroom.
The great strength of this report is the wealth of resources
that it references and the common themes that emerge across all of the
scientific disciplines. A lot of attention
is paid to the power of interactive lectures. Given that most science and
mathematics instruction is still given in traditional lecture settings, finding
ways of engaging students and getting them to think about the mathematics while they are in class is essential
for increasing student understanding.
The recommendations of effective practice range from simple
techniques, such as starting each class with a challenging question for
students to keep in mind, to transformative practices such as collaborative
learning. A common intermediate practice involves student engagement by posing
a challenging question, having students interact with their peers to think
through the answer, and then testing the answer. In some respects, this is more
easily done in the sciences where student predictions can be verified or
falsified experimentally. Yet it is also a very effective tool in mathematics
education where a well-chosen example can falsify an invalid expectation and
careful analysis can support correct understanding. But most important is that it forces to try to
use what they have been learning.
In large classes, this type of peer instruction can be
facilitated by the use of clickers. The report does include the caveat, with
supporting research, that merely using clickers without attention to how they
are used is of no measurable benefit.
The greatest learning gains that have been documented occur
when collaborative research is incorporated into the classroom. The NRC report
includes many descriptions of how this can be accomplished in a variety of
scientific disciplines. It also references the research that has established
its effectiveness. Again, attention to how it is done is an important component
of effective practice.
Two of the areas that are identified as needing more
research are issues of transference (see my September column on Teaching and
Learning for Transference) and metacognition. Usefully, the authors point
out that there are two sides to transference: the ability to draw on prior
knowledge and the ability to carry what is currently being learned to future
situations. Metacognition is an important issue in research in undergraduate
mathematics education, especially for those studying the difference between
experts and novices engaged in activities such as constructing proofs. Experts
monitor their assumptions and progress and are prepared to change track when a particular
approach is not fruitful. Novices are more likely to choose what to them seems
the likeliest approach and then ignore alternatives.
In sum, this is a useful and thought-provoking report. I
wish that it had included undergraduate mathematics education research, but
perhaps that omission can be corrected as we move forward.
Thursday, November 1, 2012
One of the goals of the MAA Calculus Study, Characteristics
of Successful Programs in College Calculus, was to gather information about the instructors of mainstream
Calculus I. Here, stratified by type of institution, is some of what we have
learned, refining some of the data presented in "The Calculus I
Instructor" (Launchings, June 2011).
Again, I am using Research University as code for institutions for which the
highest mathematics degree that is offered is the PhD, Masters University if the
highest degree is a Master's, Undergraduate College if it is a Bachelor's, and
Two Year College if it is an Associate's degree. These surveys were completed
by 360 instructors at research universities, 73 at masters universities, 118 at
undergraduate colleges, and 112 at two year colleges.
Calculus I instructors are predominantly white and male.
Masters universities have the largest percentage of Black instructors, research
universities of Asian instructors, and two-year colleges of Hispanic
instructors. By and large, undergraduate colleges do not do well in
representing any of these groups.
There is a dramatic difference between the status and
highest degree of Calculus I instructors at research universities and those at
other types of colleges and universities. At research universities, instructors
are less likely to be tenured or on tenure track, or to hold a PhD. They are
also less likely to want to teach calculus: One in five has no interest or only
a mild interest in teaching calculus. The high number of part-time faculty at
masters universities and two year colleges is troubling because of the evidence
that such instructors tend to be less effective in the classroom and much less
accessible to their students [1]. Not surprisingly, less than a quarter of the
Calculus I instructors at two-year colleges hold a PhD.
Generally, calculus instructors consider themselves to be somewhat
traditional in their instructional approaches, and they believe that students
learn best from lectures. The greatest divergence from these views is at
undergraduate colleges where almost half consider themselves to be innovative
and 45% disagree that lectures are the best way to teach. The greatest variation
among faculty at different types of institutions is over the use of calculators
on exams. Close to half of the instructors at research universities do not
allow them; 71% of the instructors at two year colleges do.
There also are institutional differences in beliefs about
whether all of the students who enter Calculus I are capable of learning this
material.
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If your child uses our Singapore Maths books for 11th Grade And 12th Grade / GCE A Level, he will be able to:
Pure mathematics
Functions and graphs
understand the terms function, domain, range and one-one function
find composite functions and inverses of functions, including conditions for their existence
understand and use the relation (fg)-1 = g-1f-1 where appropriate
illustrate in graphical terms the relation between a one-one function and its inverse
understand the relationship between a graph and an associated algebraic equation, and in particular show familiarity with the forms of the graphs of:
y = kxn where n is a positive or negative integer or a simple rational number
ax + by = c
(knowledge of geometrical properties of conics is not required)
understand and use the relationships between the graphs of y = f(x), y = af(x), y = f(x) + a, y = f(x+a), y = f(ax), where a is a constant, and express the transformations involved in terms of translations, reflections and scalings
relate the equation of a graph to its symmetries
understand, and use in simple cases, the expression of the coordinates of a point on a curve in terms of a parameter
Partial fractions
recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than:
(ax + b)(cx + d)(ex + f)
(ax + b)(cx + d)2
(ax + b)(x2 + c2)
including cases where the degree of the numerator exceeds that of the denominator
Inequalities; the modulus function
find the solution set of inequalities that are reducible to the form f(x) > 0, where f(x) can be factorised, and illustrate such solutions graphically
understand the meaning of |x| and sketch the graph of functions of the form y = lax+ b|
use relations such as lx - al < b a - b < x < a + b and |al = |b| a2 = b2 in the course of solving equations and inequalities
Logarithmic and exponential functions
recall and use the laws of logarithms (including change of base) and sketch graphs of simple logarithmic and exponential functions
recall and use the definition ax = ex ln a
use logarithms to solve equations reducible to the form ax = b, and similar inequalities
Sequences and series
understand the idea of a sequence of terms, and use notations such as un to denote the nth term of a sequence
recognise arithmetic and geometric progressions
use formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions
recall the condition for convergence of a geometric series, and use the formula for the sum to infinity of a convergent geometric series
use notation
use the binomial theorem to expand (a + b)n, where n is a positive integer
use the binomial theorem to expand (1 + x)n, where n is rational, and recall the condition |x|< 1 for the validity of this expansion
recognise and use the notations n! (with 0! = 1) and
Permutations and combinations
understand the terms 'permutation' and 'combination'
solve problems involving arrangements (of objects in a line or in a circle), including those involving
repetition (e.g. the number of ways of arranging the letters of the word NEEDLESS)
restriction (e.g. the number of ways several people can stand in a line if 2 particular people must - or must not - stand next to each other)
Trigonometry
use the sine and cosine formulae
calculate the angle between a line and a plane, the angle between two planes, and the angle between two skew lines in simple cases
Trigonometrical functions
understand the definition of the six trigonometrical functions for angles of any magnitude
recall and use the exact values of trigonometrical functions of
use the notations sin-1x, cos-1x and tan-1x to denote the principal values of the inverse trigonometrical relations
relate the periodicity and symmetries of the sine, cosine and tangent functions to the form of their graphs, and use the concepts of periodicity and / or symmetry in relation to these functions and their inverses
use trigonometrical identities for the simplification and exact evaluation of expressions, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of:
and equivalent statements
the expansion of sin(A ± B), cos(A ± B) and tan(A + B)
the formulae for sin2A, cos2A and tan2A
the formulae for sin A ± sin B and cos A ± cos B
the expression of a in the forms
find the general solution of simple trigonometrical equations, including graphical interpretation
use the small-angle approximations
Differentiation
understand the idea of a limit and the derivative defined as a limit, including geometrical interpretation in terms of the gradient of a curve at a point as the limit of the gradient of a suitable sequence of chords
recognise when an integrand can usefully be regarded as a product, and use integration by parts to integrate, e.g., x sin 2x, x2ex, ln x
use the in method of integration by substitution to simplify and evaluate either a definite or an indefinite integral (including simple cases in which the candidates have to select the substitution themselves, e.g. )
evaluate definite integrals (including e.g.
understand the idea of the area under a curve as the limit of a sum of the areas of rectangles and use simple applications of this idea
use integration to find plane areas and volumes of revolution in simple cases
use the trapezium rule to estimate the values of definite integrals, and identify the sign of the error in simple cases by graphical considerations
Vectors
use rectangular cartesian coordinates to locate points in three dimensions, and use standard notations for vectors, i.e. xi + yj + zk, a
carry out addition and subtraction of vectors and multiplication of a vector by a scalar, and interpret these operations in geometrical terms
use unit vectors, position vectors and displacement vectors
recall the definition of and calculate the magnitude of a vector and the scalar product of two vectors
use the scalar product to determine the angle between two directions and to solve problems concerning perpendicularity of vectors
understand the significance of all the symbols used when the equation of a straight line is expressed in either of the forms r = a + tb and and convert equations of lines from vector to Cartesian form and vice versa
solve simple problems involving finding and using either form of the equation of a line
use equations of lines to solve problems concerning distances, angles and intersections, and in particular:
determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists
find the perpendicular distance from a point to a line
find the angle between two lines
use the ratio theorem in geometrical applications
Mathematical induction
understand the steps needed to carry out a proof by the method of induction
use the method of mathematical induction to establish a given result e.g. the sum of a finite series, or the form of an nth derivative
Complex numbers
understand the idea of a complex number, recall the meaning of the terms 'real part', 'imaginary part', 'modulus', 'argument', 'conjugate', and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal
carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in cartesian form (x + iy)
recall and use the relation zz* = |z|2
use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs
represent complex numbers geometrically by means of an Argand diagram
carry out operations of multiplication and division of two complex numbers expressed in polar form ()
understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying, dividing two complex numbers
Curve sketching
determine, in simple cases, the equations of asymptotes parallel to the axes
use the equation of a curve, in simple cases, to make deductions concerning symmetry or concerning any restrictions on the possible values of x and / or y that there may be
sketch curves of the form y = f(x), y2 = f(x) or y = |f(x)| (detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as turning points, asymptotes and intersections with the axes)
First order differential fquations
formulate a simple statement involving a rate of change as a differential equation, including the introduction if necessary of a constant of proportionality
find by integration a general form of solution for a first order differential equation in which the variables are separable
find the general solution of a first order linear differential equation by means of an integrating factor
reduce a given first order differential equation to one in which the variables are separable or to one which is linear by means of a given simple substitution
understand that the general solution of a differential equation is represented in graphical terms by a family of curves, and sketch typical members of a family in simple cases
use an initial condition to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation
Numerical methods
locate approximately a root of an equation by means of graphical considerations and / or searching for a sign change
use the method of linear interpolation to find an approximation to a root of an equation
understand the idea of, and use the notation for, a sequence of approximations which converges to the root of an equation
understand how a given simple iterative formula of the form relates to the equation being solved, and use a given iteration to determine a root to a prescribed degree of accuracy (conditions for convergence are not included)
understand, in geometrical terms, the working of the Newton-Raphson method, and derive and use iterations based on this method
appreciate that an iterative method may fail to converge to the required root
Particle mechanics
Forces and equilibrium
identify the forces acting in a given situation
understand the representation of forces by vectors, and find and use resultants and components
solve problems concerning the equilibrium of a particle under the action of coplanar forces (using equations obtained by resolving the forces, or by using a triangle or polygon of forces)
recall that the contact force between two surfaces can be represented by two components (the 'normal component' and the 'frictional component') and use this representation in solving problems
use the model of a 'smooth' contact and understand the limitations of this model
understand the concept of limiting friction and limiting equilibrium, recall the definition of coefficient of friction; and use the relationship as appropriate (knowledge of angle of friction will not be required)
recall and use Newton's third law
Kinematics of motion in a straight line
understand the concepts of distance and speed, as scalar quantities, and of displacement, velocity and acceleration, as vector quantities, and understand the relationships between them
sketch and interpret x-t and v-t graphs, and in particular understand and use the facts that:
the area under a v-t graph represents displacement
the gradient of an x-t graph represents velocity
the gradient of a v-t graph represents acceleration
use appropriate formulae for motion with constant acceleration in a straight line
Newton's laws of motion
recall and use Newton's first and second laws of motion
apply Newton's laws to the linear motion of a particle of constant mass moving under the action of constant forces (including friction)
solve problems on the motion of two particles, connected by a light inextensible string which may pass over a fixed smooth light pulley or peg
Energy, work and power
understand the concept of the work done by a force, and calculate the work done by a constant force when its point of application undergoes a displacement not necessarily parallel to the force (use of the scalar product is not required)
understand the concepts of gravitational potential energy and kinetic energy, and recall and use appropriate formulae
understand and use the relationship between the change in energy of a system and the work done by the external forces, and use where appropriate the principle of conservation of energy
recall and use the definition of power as the rate at which a force does work, and use the relationship between power, force and velocity for a force acting in the direction of motion
solve problems involving, for example, the instantaneous acceleration of a car moving on a hill with resistance
Linear motion under a variable force
solve simple problems on the linear motion of a particle of constant mass moving under the action of variable forces by setting up and solving an appropriate differential equation (use of for velocity and , as appropriate, for acceleration is expected, and any differential equations to be solved will be first order with separable variables)
Motion of a projectile
model the motion of a projectile as a particle moving with constant acceleration, and understand the limitations of this model
use horizontal and vertical equations of motion to solve problems on the motion of projectiles (including finding the magnitude and direction of the velocity at a given time or position and finding the range on a horizontal plane)
derive and use the cartesian equation of the trajectory of a projectile, including cases where the initial speed and/or angle of projection is unknown (knowledge of the range on an inclined plane is not required)
Hooke's law
recall and use Hooke's law as a model relating the force in an elastic string or spring to the extension or compression, and understand and use the term 'modulus of elasticity'
understand the concept of elastic potential energy, and recall and use the appropriate formula for its calculation
use considerations of work and energy to solve problems involving elastic strings and springs
Uniform circular motion
understand the concept of angular speed for a particle moving in a circle with constant speed, and recall and use the relation (no proof required)
understand that the acceleration of particle moving in a circle with constant speed is directed towards the centre of the circle and has magnitude (no proof required)
use Newton's second law to solve problems which can be modelled as the motion of a particle moving in a circle with constant speed
Probability and statistics
Probability
use addition and multiplication of probabilities, as appropriate, in simple cases, and understand the representation of events by means of tree diagrams
understand the meaning of mutually exclusive and independent events, and calculate and use conditional probabilities in simple cases
understand and use the notations P(A), P(A|B) and the equations and = P(A) P(B|A) = P(B) P(A|B) (the general form of Bayes' theorem is not required)
Discrete random variables
understand the concept of a discrete random variable
construct a probability distribution table relating to a given situation, and calculate E(X) and Var(X)
appreciate conditions under which a uniform distribution or a binomial distribution B(n,p) may be a suitable probability model, and recall and use formulae for the calculation of binomial probabilities
understand conditions under which a Poisson distribution Po() may be a suitable probability model, and recall and use the formula for the calculation of Poisson probabilities
recall and use the means and variances of binomial and Poisson distributions
use a Poisson distribution as an approximation to a binomial distribution, where appropriate (candidates should know that the conditions n > 50 and np < 5, approximately, can generally be taken to be suitable)
The normal distribution
recall the general shape of a normal curve, and understand how the shape and location of the distribution are affected by the values of (in general terms only; no knowledge of mathematical properties of the normal density function is included)
standardise a normal variable and use normal distribution tables
use the normal distribution as a probability model, where appropriate, and solve problems concerning a variable X, where , including:
finding the value of P(X < x1) given the values of
use of the symmetry of the normal distribution
finding a relationship between given the value of P(X < x1)
repeated application of the above
recall conditions under which a normal distribution may be used to approximate a binomial distribution (n sufficiently large to ensure that np > 5 and nq > 5. approximately) or Poisson distribution (, approximately), and calculate such approximations, including the use of a continuity correction
Samples
understand the distinction between a sample and a population, and appreciate the necessity for randomness in choosing samples
explain in simple terms why a given sampling method may be unsatisfactory (a detailed knowledge of sampling and survey methods is not required)
recognise that the sample mean can be regarded as a random variable, and use the facts that
use the fact that X is normally distributed if X is normally distributed
use the Central Limit Theorem (without proof) to treat X as being normally distributed when the sample size is sufficiently large ('large' samples will usually be of size at least 50, but candidates should know that using the approximation of normality can sometimes be useful with samples that are smaller than this)
calculate unbiased estimates of the population mean and population variance from a sample (only a simple understanding of the term 'unbiased' is required)
determine, from a sample from a normal distribution of known variance or from a large sample, a confidence interval for the population mean
determine, from a large sample, a confidence interval for a population proportion
Linear combinations of random variables
recall and use the results in the course of solving problems that, for either discrete or continuous random variables,
E(aX + b) = aE(X) + b and Var(aX + b) = a2 Var(X)
E(aX + bY) = aE(X) + bE(Y)
Var(aX + bY) = a2 Var(X) + b2 for independent X and Y
recall and use the results that:
if X has a normal distribution then so does aX + b
if X and Y have independent normal distributions then aX + bY has a normal distribution
if X and Y have independent Poisson distributions then X + Y has a Poisson distribution
Continuous random variables
understand and use the concept of a probability density function, and recall and use the properties of a density function (which may be defined 'piecewise')
use a given probability density function to calculate the mean, mode and variance of a distribution, and in general use the result in simple cases, where f(x) is the probability density function of X and g(X) is a function of X
understand and use the relationship between the probability density function and the distribution function and use either to evaluate the median, quartiles and other percentiles
use a probability density function or a distribution function in the context of a model, including in particular the continuous uniform (rectangular) distribution
Hypothesis testing
understand and use the concepts of hypothesis (null and alternative), test statistic, significance level, and hypothesis test (1-tail and 2-tail)
formulate hypotheses and apply a hypothesis test concerning the population mean using:
a sample drawn from a normal distribution of known variance
a large sample drawn from any distribution of unknown variance
formulate hypotheses concerning a population proportion, and apply a hypothesis test using a normal approximation to a binomial distribution
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Archive for category Mathematics
For most students, math is one subject that is very difficult and makes them frustrated. They encountered many difficulties to perform tasks that contain the various figures and symbols. However, there are several other students are very pleased with this lesson. In fact, they have a proud achievement in various events. For students who have trouble with math, they can take advantage of online math help is available on the internet. The professional staff is ready to assist the various difficulties in understanding the subject matter.
Mathematics also has a scope of discussion is very broad. Students will usually find difficulty in math in some areas such as algebra. Algebra is one part of the subject matter of mathematics associated with the various calculations and formulas. In this case, if students find it difficult to find the solution of tasks of linear algebra, which he acquired at school, they can rely on online services for solving linear algebra online so students can do some asking to get answers from such difficulties.
In addition to algebra, other fields are also associated with numbers of calculus. Students will usually feel tired when you get this lesson in the school because almost all of the discussion in this section contains figures that have various meanings and functions. If a student wants to get deeper assistance about calculus course materials or to ask about the various tasks that he obtained, students can take advantage of online calculus help. The service is very helpful for students to learn better and more enjoyable.
The concept of function is very important part of Calculus because of its close relation with various phenomena of reality. Before going through the formal definition of function, it is better to understand it through some illustrated examples.
Suppose that a particle is moving at the uniform rate 3 kilometers per hour. If S denotes the distance in kilometers and T time in hours, then S = distance covered in T hours = 3T
Thus, it is clear that when T changes, S also changes and corresponding to each non negative value of T, there will be only one value of S. In this situation we say that S is a function of T. Here, T is an independent variable over the set of non negative real numbers because T can be any arbitrary non negative real number. S is a dependent variable as its value depends on the value of T. Here S also varies over the set of non negative real numbers. We will call the set over which independent variable T varies the domain and the set over which S varies the range of the function. Here domain and range of the function are both a set of non negative real numbers.
Now we coming to the formal definition of function – Let A and B be two non-empty sets, then a rule 'f' which associates with the each element of A with a unique element of B is called a mapping or function from A into B. If f is a mapping from A into B we write f: A → B (read as f is a mapping from A into B). If f associates x Є A to y Є B, then we say that y is the image of the element x under the map f and denote it by f(x) and we write y = f(x). The element x is called as pre-image or inverse-image of y.
Function in calculus is one of the scoring topics in a math exam. If you are facing problem understanding these topics then don't get nervous. It is one of the common challenges faced by many students these days. You just need the help of expert private calculus tutor who can guide you through the concepts. Spend some time online and find the right private calculus tutor in your locality to get over your calculus fear.
If you are facing problems solving calculus papers then you can take the help of expert private calculus tutor. It is now possible to find the right tutor in your locality through Internet.
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Heya guys! Is anyone here know about glencoe algebra 1 workbook? I have this set of problems regarding it that I just can't understand. Our class was tasked to solve it and understand how we came up with the answer . Our Algebra professor will select random students to answer it as well as explain it to class so I require comprehensive explanation regarding glencoe algebra 1 workbook. I tried answering some of the questions but I guess I got it completely wrong . Please help me because it's a bit urgent and the deadline is quite close already and I haven't yet understood how to solve this.
Hey friend ! Studying glencoe algebra 1 workbook online can be a nightmare if you are not a pro at it. I wasn't an expert either and really regretted my decision until I found Algebrator. This little program has been my buddy since then. I'm easily able to solve the problems now.
I am a regular user of Algebrator. It not only helps me complete my assignments faster, the detailed explanations offered makes understanding the concepts easier. I strongly recommend using it to help improve problem solving skills.
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Tag Archives: Geogebra
STEM (Science, Technology, Engineering, and Mathematics) curriculum is focused on student problem-solving, discovery, exploratory learning. It requires students to actively engage in a situation in order to find its solution.
Teq's STEM Middle School and High School course will explore how SMART Technologies, and programs like Google Sketchup, Algodoo, and Geogebra, can enable students to design, … Read more
Teq's Summer Math Software Course will challenge your view of what's possible on a SMART Board. We will go beyond SMART Notebook Math Tools and take a look at some of the best math software available. Our focus will be on a student-centered classroom, and we'll explore programs such as Logger Pro, Fluid Math, Geometer's Sketchpad, TI-SmartView, Geogebra, Google … Read more
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Helpful Hints for Mathematics: * Work a few problems every night * Study for quizzes and tests by practicing homework problems * Review the student agenda daily for class assignments * Make sure to get enough to eat and sleep
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Algebra 2 with Trigonometry
The AskDrCallahan Algebra II with Trig (sometimes called Pre-calculus) will develop your son's or daughter's skill in math and prepare them future courses in math, the ACT and SAT, and specifically college or high school calculus.
This course is taught like we would teach it in the college setting, so the student will not only be getting a taste of the content, but also the pace and treatment of such a math course at the college level.
Algebra II with Trig DVD Set Videos of course content - Approximately 14 hours of video following the textbook. Includes the tests, the test grading guide, and the Syllabus.
Disk 1 - Contains the Teachers Guide and the Solutions to Selected Problems. The Solutions to Selected Problems provides solutions to all the problems we have assigned on our syllabus which are not clearly answered in the back of the textbook. Both documents are PDF files to be used on your computer. They can be printed as needed. The same files can be downloaded at the links below.
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Of course, problem-solving is a major activity in any mathematics
course. A sound approach to the problem-solving process is a
necessity for mathematics students. The approach outlined here is
essentially the one from the classic book on the subject, How To
Solve It by George Pólya1.
The first step is understanding the problem. The student must be
able to state what needs to be solved, and what supporting conditions
are given as information to be used in solving the problem. Once these
are understood, it can be helpful (when possible) to draw a picture
representing the unknown quantity to be solved and the other given
information. If the unknown quantities are given in verbal form, it is
necessary to introduce a suitable notation for these variables, and
for the given conditions. Once we understand what we are given, and
what we are looking for, we can proceed to the next step.
We need to devise a plan for the solution of the problem. This plan
should arise from a connection between the given information and the
unknown. If an immediate connection isn't apparent, the student may
have to explore other auxiliary connections to develop a chain which
will link the given information to the unknown.
Once a plan has been developed, the student must carry it out. It is
especially important to check the validity and the accuracy of each
step in the plan.
Finally, the student should look back on the solution obtained and
examine it. This means checking the work, and also stepping back to
get an overview of the entire process. The solution to this problem
then becomes part of the student's problem-solving library, which
should be available for application to similar problems in the future.
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Maths
Maths at Derby
What's next?
Degree level mathematics at Derby uses the skills you've developed at A level, particularly algebra and calculus, as the building blocks for learning new concepts and mathematical techniques. You'll learn how to use these effectively to model and solve problems relevant to industry and organisations.
Maths for the real world
First year modules cover the essential fundamental techniques and methods of maths, including differential and integral calculus, solution of differential equations, complex numbers, matrices, and the use of mathematical software.
Second year modules include mathematical methods and modelling, and you'll be involved in the Maths group project for a local company. You'll apply your knowledge in innovative ways to solve a real business problem for them.
In your final year you'll complete a dissertation. You'll also study more advanced maths, statistics and operational research. Topics include genetic algorithms, tabu search, game theory, fractals, number theory and cryptography.
Professional accreditation
Our Mathematics courses are approved by the Institute of Mathematics and its Applications (IMA), which means you'll be accepted for associate membership upon graduation. This is the first step to becoming a Chartered Mathematician (C.Math).
International students
If you're an international student, see more information on how we help and support you.
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Book Description: This well-respected text gives an introduction to the theory and application of modern numerical approximation techniques for students taking a one- or two-semester course in numerical analysis. With an accessible treatment that only requires a calculus prerequisite, Burden and Faires explain how, why, and when approximation techniques can be expected to work, and why, in some situations, they fail. A wealth of examples and exercises develop students' intuition, and demonstrate the subject's practical applications to important everyday problems in math, computing, engineering, and physical science disciplines. The first book of its kind built from the ground up to serve a diverse undergraduate audience, three decades later Burden and Faires remains the definitive introduction to a vital and practical subject.
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Welcome to the 2009-2010 math homepage.
All math courses will be graded as follows:
1/3 = Homework, seatwork and projects
1/3 = Quizzes and tests
1/3 = Effort and attitude
Math classes will be held on Monday, Wednesday and Friday of each week. On Tuesdays and Thursdays, seatwork will be assigned. Seatwork is listed on the whiteboard in the math classroom. Students are expected to complete seatwork before leaving school on Tuesday and Thursday. Textbooks are not to leave the school building. Seatwork will be checked at the beginning of the next scheduled math class.
Math Class Schedule (Mon. - Wed. - Fri.)
Algebra 2: 8:15 - 9:00
Geometry: 9:30 - 10:15
Algebra 1: 10:15 - 11:00
Pre Algebra: 11:00 - 11:30
Please plan your schedule so that you are on time for every class. Attendance is part of your effort and attitude grade.
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Contains information on the following subjects: straight
lines, conic sections, tangents, normals, slopes; introduction todifferential and integral calculus; combinations and permutations;
and introduction to probability. This course is general innature and is not directed toward any specific specialty.
Assists enlisted and officer personnel of the United
States Navy and Naval Reserve in acquiring the knowledge requisite to thecomputation of time. It uses two-dimensional charts and expanded
narratives to explain both the global division anddesignation, and the processes and mathematical formulas used in
the conversion of time.
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Number Theory and Its Mathematical Structures Editions
Chegg carries several editions of the Number Theory and Its Mathematical Structures textbook.
Below you'll find a list of all the Number Theory and Its Mathematical Structures editions available to rent or buy.
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The Aftermath of Calculator Use in College Classrooms
13.11.2012
Students may rely on calculators to bypass a more holistic understanding of mathematics, says Pitt researcher
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Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center.
King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology.
"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
Together with Carol Robinson, coauthor and director of the Mathematics Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves—a mathematical function that describes a smooth repetitive oscillation—and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
King also suggests that relevant research should be done investigating the correlation between how and why students use calculators to evaluate the types of learning approaches that students adopt toward problem solving in mathematics
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...I use the book Algebra and Trigonometry with Analytic Geometry, by Swokowski and Cole, 10th edition. This is a college level book and I am confident it will help with learning the concepts of the course. High school algebra 2 requires students to dig deeper than the basic principles introduced ...Included in skill activities is written work, which involves generating and encoding text by applying all the new things we have learned. Having an MSEd in Special Education has enabled me to design curriculum to help students learn a variety of study skill techniques. In learning study skills,...
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Algebra is fundamental to the working of modern society, yet its origins are as old as the beginnings of civilization. Algebraic equations describe the laws of science, the principles of engineering, and the rules of businessAlgebra is fundamental to the working of modern society, yet its origins are as old as the beginnings of civilization. Algebraic equations describe the laws of science, the principles of engineering, and the rules of business.
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Clear, lively style covers all basics of theory and application, including mathematical models, elementary concepts of graph theory, transportation problems, connection problems, party problems, diagraphs and mathematical models, games and puzzles, graphs and social psychology, planar graphs and coloring problems, and graphs and other mathematics.
This book fills a need for a thorough introduction to graph theory that features both the understanding and writing of proofs about graphs. Verification that algorithms work is emphasized more than their complexity. An effective use of examples, and huge number of interesting exercises, demonstrate the topics of trees and distance, matchings and factors, connectivity and paths, graph coloring, edges and cycles, and planar graphs. For those who need to learn to make coherent arguments in the fields of mathematics and computer science.
This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems.
The book is intended to be an introductory text for mathematics and computer science students at the second and third year level in universities. It gives an introduction to the subject with sufficient theory for that level of student, with emphasis on algorithms and applicationsThis book written by experts in their respective fields, and covers a wide spectrum of high-interest problems across these discipline domains. The book focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineeringIn this substantial revision of a much-quoted monograph first published in 1974, Dr. Biggs aims to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first section, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth.
This is a highly self-contained book about algebraic graph theory which iswritten with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvaluesReviewing recent advances in the Edge Coloring Problem, Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture provides an overview of the current state of the science, explaining the interconnections among the results obtained from important graph theory studies. The authors introduce many new improved proofs of known results to identify and point to possible solutions for open problems in edge coloring.
Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances.
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Temporarily out of stock
IGCSE Mathematics for CIE (International GCSE)
Innovative - Interactive - InternationalCollins IGCSE Mathematics has been developed to give maximum support for students studying for the Cambridge International Examinations GCSE. International examples are used throughout, 'localising' learning. An interactive CD ROM, supporting study and revision, is included.
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YourTeacher has aligned our lessons to over 160 textbooks in middle school, high school, community college and college. By simply entering the page number you are working on, YourTeacher will return the exact lesson covering that page of your textbook! Our textbook search tool is particularly helpful for math homework help, where students often get stuck and need background instruction in order to move forward.
Need Online Math Help? Grade Reporting and Progress Tracking
YourTeacher provides individualized grade reporting and progress tracking to ensure that parents, administrators and students know if the material is being mastered and how much the content is being utilized.
Grade Reporting
Each lesson contains a multiple-choice self-test to prove mastery. Unlike other programs, YourTeacher's self-tests can be taken multiple times with new problems each time. This allows students to continue to take self-tests until mastery has been proven.
The results of the self-tests are recorded in the grade management system. The grades are a simple way for parents, administrators, and students to ensure that students are receiving the online math help they need and as a way to identify areas of weakness. Final grades for entire courses are also available online.
Progress Tracking
In addition to our grade reporting system, YourTeacher keeps track of student progress by monitoring usage. Parents, administrators, and students are able to see the number of lessons completed, the number of incomplete self-tests, total logins, and the last login. These basic metrics ensure that student usage can be easily tracked.
YourTeacher uses the results of self-tests to automatically recommend background lessons based on areas of weakness. As an example, if a student scored below an 80% on a lesson such as Comparing Proper Fractions (Pre-Algebra), our system would automatically recommend the background lessons required to master this lesson. In this example, our system would automatically recommend the following lessons: Comparing Numbers, Multiples and Least Common Multiples, Equivalent Fractions, and Introduction to Fractions.
When these background lessons are presented, students can also quickly see which of these lessons they have already mastered and which they need to work on. After completing the background lessons, the student then can re-take the self-test for Comparing Proper Fractions. Because the self-test is different each time, students will have to prove true mastery. If they score an 80% or above, the lesson is automatically removed from the recommendation list (note that if a student scores less than an 80% on one of the background lessons, then the system will continue to recommend lessons going backward until the student 'bottoms out').
This adaptive recommendation system ensures that no matter where a student starts, they can get the individualized online math help they need to succeed.
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Problems in Mathematics
Aims
This course aims to allow students to engage with some of the important problems which have shaped mathematics.
Problems will be put in their historical context and will be used to illustrate the development of different areas of mathematics.
You will have the opportunity to tackle more open-ended work and make links between the many branches of mathematics that have been studied on the degree programme.
Teaching and Assessment
This module is entirely coursework based; split into 40% for problem sheets and 60% for essays. Over the course of the year there will be several evenings of lectures. Each evening will concentrate on one topic (the choice of topics will vary each year). At the end of each lecture evening, students will be given a problem sheet to complete. This will consist of several short compulsory questions to be submitted within 4 weeks of the lecture.
Each problem sheet will count 10% towards the final mark for the module, and students will complete four problem sheets.
In addition, at the end of each lecture students will be given a short list of suggestions for essays with each topic.
Over the course of the year students must choose any two of these questions to complete; each essay should be roughly 2,500 words and no more than 4,000 words.
Optionally a student may, with permission, choose to write ONE essay of the two on a mathematical subject of their own choosing. If a student wishes to do this, he or she must obtain the permission of a member of School to supervise this project, and submit an abstract which must be approved by the essay supervisor before the end of the Spring Term.
Each essay counts 30% giving a total of 60%.
Syllabus
A selection of typical topics is given below but will vary from year to year to keep current. Each topic would be the subject of one evening of lectures.
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Fostering Algebraic Thinking
Mark Driscoll's Fostering Algebraic Thinking School-Based Seminar expands on his popular book and professional-learning toolkits, leading participants to in-depth knowledge of the key ideas underpinning algebraic thinking.
Teachers will explore algebraic thinking from two powerful perspectives. First they solve open-ended problems and observe their own algebraic habits of mind. Then they analyze student work to see how kids approached the problem. Ultimately they will discover commonalities and find out how knowledge of both approaches can inform instruction.
During this seminar teachers will deepen their understanding of algebraic thinking through hands-on investigation and discussions and analysis of student work. In addition, they will learn structured approaches for analyzing student work that distinguish between evidence and interpretation, and reflect on ways to elicit productive algebraic thinking from students
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Quantitative Reasoning
QR 100 Basic Quantitative Reasoning 3 Prereq.: Permission of instructor or department chair. Designed to improve student's ability to succeed in mathematics courses and other disciplines requiring quantitative reasoning, problem-solving skills and overcoming math anxiety. Students will be given diagnostic tests to identify areas requiring remediation and will take the mathematics placement examination at the end of the course. This does not meet the prerequisite for any mathematics course and may not be used to meet the general education requirement or any major or minor in mathematics.
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Resources
Mathematics SAP
Mathematics can be roughly divided into two basic fields — theoretical mathematics and applied mathematics. Practitioners of applied mathematics solve problems in various fields using mathematical models. Practitioners of theoretical mathematics investigate mathematical conjectures and construct proofs of their validity or find counter examples to prove their falsity. In either case, mathematics majors must be able to communicate verbally and in written form the solutions of problems, proofs of theorems, or counter-examples. As mathematics incorporates the use of symbolic reasoning involving, not surprisingly, the use of symbols not a part of common language, practitioners must be able to use communication technology to create a record of the results of their learning endeavors. The Student-as-Practitioner components in mathematics courses are designed to promote that communication.
Mathematics majors at Lakeland are asked to submit solutions and proofs in either written form or as oral presentations. In general, freshman and sophomore courses instructors allow handwritten solutions and proofs for submitted work. In junior- and senior-level courses the use of a word processor with an equation editor is required for submitting work. In addition, students in College Geometry (MAT 322) are required to write an expository term paper containing both historical information as well as mathematical content.
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, intended for a graphing calculator optional college algebra and trigonometry course, offers students the content and tools they will need to successfully master college algebra and trigonometry. The authors have addressed the needs of students who will continue their study of mathematics, as well as those who are taking college algebra and trigonometry as their final mathematics course. Emphasis is placed on exploring mathematical concepts by using real data, current applications and optional technology.
Oblique Triangles and the Law of Sines. The Law of Cosines. Vectors and Their Applications. Products and Quotients of Complex Numbers. Powers and Roots of Complex Numbers. Polar Equations. Parametric Equations.
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{"itemData":[{"priceBreaksMAP":null,"buyingPrice":11.69,"ASIN":"0486277097","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":3.99,"ASIN":"0486270785","isPreorder":0}],"shippingId":"0486277097::T2kU%2FutiwHe0Kgd9%2BqQwMcxxGzzZTv3BhvpHZFtLGkVF41wy5gt7mVaz1Gu49gET5RoYgncdqwb72D%2FIKvAkykVuAmQZt2SFTVBnpD7NIWk%3D,0486270785::oKWzOfHt82Tii4kBfxDKKlBfLYSXYgds0v4TxCfIZihKnxCT5LCpOcyjMTVGJttPZ6nDFPH8iRDtoTWKwWtNVTStAx1sD%2BJTD60k5ojmK "320 unconventional problems in algebra, arithmetic, elementary number theory and trigonometry." The problems are mathematically accessible to students at the high school level or higher, as they call more upon analytical thinking than upon advanced mathematical techniques. There is a range of difficulties, with harder problems marked with stars in the book. (The hardest problems are marked with double stars.) The problems are divided into twelve sections: "Introductory Problems," "Alterations of Digits in Integers," "The Divisibility of Integers," "Some Problems from Arithmetic," "Equations Having Integer Solutions," "Evaluating Sums and Products," "Miscellaneous Problems from Algebra," "The Algebra of Polynomials," "Complex Numbers," "Some Problems of Number Theory," "Some Distinctive Inequalities," and "Difference Sequences and Sums." Much of the book is devoted to providing hints and solutions, which are both thorough and clear. This is a great resource for preparing for competition, for developing your analytical thinking, or just for having fun (that is, if you are the sort of person who finds solving math problems fun).
The reason I am giving 5 stars to this book is for its unique collection of problems. It has been very entertaining reading the book so far (I have not completed the book). There however are a few errors which can be easily figured out by the reader. The treatment of each problem is unique.
This is a great resource for challenging math problems. After getting annoyed with newspaper "problem of the week" type books, this was a refreshing find. Don't let the "high school level" disclaimer fool you - there are some seriously difficult problems in here.
If you're the type to find logic and math problems fun, I would recommend dropping $15 for this text. It's well worth the time.
I found this book very interesting, because it deals with many many problems of algebra and number theory. You can find many interesting and tough theorems (not all of them are widely known nor taught) with their demonstration. I particularily liked the section about "distinctive inequalities": it deals with a great number of inequalities and you can learn some new techniques for solving them. The book lacks of geometry, that's true (only some trig somewhere), but it gives (in my opinion) really a strong preparation on topics concerning algebra... try it!
This books is what every book on math olympiads should be, it deals with high level problems in a way that readers can easily follow; I also liked it because there are some problems who have many interesting solutions and generalizations
It is a problem and solution type of book. The organization is not ideal. The problems are not ordered by difficulties. There are some very difficult problems. The solutions are quite rigorous and mostly well explained. Probably nice for a teacher to use, not good for self study.
Almost all the problems are for algebra, which seems narrow for preparation for high school math Olympiad. From the forward, it is stated that they are for 8th and 9th graders in USSR, which probably explain the lack of Euclidean geometry.
If you like challenging math problems then this book is for you. It's ideal if you want to prepare for a national math olympiad or if you just like hard math problems. Get this book and you won't regret it !!!
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ADVANCED ALGEBRA LESSONS
Whether needing help with advanced algebra homework or reviewing for tests, Mr. X can help math students better understand Advanced Algebra. Our lessons are designed to reinforce the instructor's message. We also have a library of sample algebra problems with examples of solved problems for each advanced algebra lesson. Check out our free samples below, as well as the advanced algebra curriculum. Advanced algebra lessons and problems are included with a subscription to Mr. X.
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Designed for use by those in Years 7/8, "So You Really Want to Learn Maths 3" is the ideal resource for Key Stage 3 mathematics pupils as well as those working towards papers 2 and 4 of the 13+ Common Entrance examination or scholarship. The So you really want to learn Maths course is a rigorous, thorough mathematical course for those who really want to learn. Clear explanations are followed by an impressive amount of practice exercises which will ensure that even the fastest mathematicians will never run out of exercises! This course is ideal for use at school or by parents and home schoolers looking for a textbook which takes a rigorous approach to mathematics whilst also providing clear explanations of current mathematical methods and plenty of exercises for consolidation. An accompanying Answer Book is also available to purchase separately.
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Mathematics Framework Solution Sets
Solution sets for the sample standards-aligned problems listed in Appendix D of the Mathematics Framework.
Solution sets for the sample standards-aligned problems listed in Appendix D of
the Mathematics Framework for grades five, six, seven, and the disciplines of
Algebra I and Geometry. Solution sets are intended for teacher use, not student
use. They represent one way of solving the problems but are not intended to
represent the only way of solving the problems.
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Bell Gardens CalculusHowever, when more than three equations need to be solved simultaneously, one needs to begin using arrays and matrices. Thus, central to linear algebra is the study of matrices and how to perform basic operation such as matrix multiplication. The notion of vector space and subspace becomes important and Eigenvalue problems will be introduced in more advanced linear algebra courses.
...Finally, many teaching techniques are effective some students; meanwhile, they aren?t efficient with others. It?s our responsibility to determine a useful teaching method so that we can guide a student appropriately. In addition, we should never discourage students from learning by saying negative words under any circumstances
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Standard Deviants is a series of edutainment videos produced in the United States dealing with educational subjects such as math, science, politics, English, social studies, foreign language, and SAT Test Prep created for elementary schoolers, middle schoolers, high schoolers, college students, and graduate school students in the U.S. Originally a Public Broadcasting Service (PBS) Public television series, it blends essential information with Saturday Night Live-style humor. It is now a division of Cerebellum Corporation. The show is divided into many sections, using a humorous entrance to each one. An example of this is an Algebra subplot where Idaho Bones, based on Indiana Jones goes on a quest to find the Golden X.
The Twisted World of Trigonometry. The DVD Super Pack contains Modules 1 through 6.
Programs Included in this Series:
Module 1: The Basics
The Standard Deviants serve-up all sorts of useful trig vocabulary. Get your fill of degree, radian measurements and a sampler platter of right triangle trigonometry.
Module 2: Trigonometry Functions
The Standard Deviants take you to the junction of all functions. Together, we'll learn about six trigonometry functions.
Module 3: Triangles
It's time to clean out your brain to make room for triangles! The Standard Deviants cover lots of material including: right triangles, oblique triangles, the law of sines and the law of cosines.
Module 4: Graphing Functions
The Standard Deviants get graphic. It's off to the world of x and y axes, origins and amplitude. Learn some helpful rules to make your trig gig much easier. The Standard Deviants even hook you up with some key trig formulas.
Module 5: Identities
Discover the amazing secrets behind identities! The Standard Deviants start with a quick look at some common trig graphs which can be lines, curves or even parts of a circle. Then they plow headfirst into identities and the formulas that you'll need to make them happen.
Module 6: Angle Formulas
It's time to angle yourself for some trig learning because the Standard Deviants are here to discuss angle formulas, identities and proofs. And no trig experience would be complete without examples, lots and lots of examples to help solidify your trig knowledge
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0534462790
9780534462796 students too often leave a precalculus class unprepared to go on to calculus. Although students who complete a precalculus course generally have had plenty of algebra and trigonometry review, they often lack the grounding in analysis and graphing necessary to make the transition to calculus. Faires and DeFranza's PRECALCULUS concentrates on teaching the essentials of what a student needs to fulfill their precalculus requirement and to fully prepare them to succeed in calculus. This streamlined text provides all the mathematics that students need--it doesn't bog them down in review, or overwhelm them with too much, too soon. And the authors have been careful to keep this book, unlike many of the precalculus books on the market, at a length that can be covered in one term. «Show less... Show more»
Rent Precalculus (with BCA/iLrn Tutorial and InfoTrac) 3rd Edition today, or search our site for other Faires
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Mathcad 14 Inc.Crack System PTC Mathcad 14 - provides a powerful, convenient and intuitive way of describing the algorithms for solving mathematical problems.
MathCAD system is so flexible and versatile, that can provide invaluable assistance in solving mathematical problems as a student, master the basics of mathematics, and Academician, working with complex scientific problems.
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Patterns - Numbers, Shapes, etc. (Lesson 14 of 61 math test prep lesson that covers patterns that can be created by numbers and shapes and the algebraic expressions that can represent those patterns, including Pascal's triangle, as part of the Algebra
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
2693.48
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Note: Don't look at the 2010 exams! Your school will most likely use them as your practice exams and you'll get the best understanding of what you need to revise if you take those exams without having seen them beforehand. Just like the real thing! If you want your best chance of passing further maths, don't look at the 2010 past exams until well after your practice exam period in around September/October.
VCE Notes and Forum
This site has some further notes for download, but the best bit is the student forum – current VCE students answering questions for other current VCE students. Worth a look.
Essential Further Maths Extras
The authors of the Essential Further maths textbook (Jones, Evans and Lipson) have created a series of quick presentations and activities on most of the topics covered in the book. They sum up the theory for each concept nicely, and watching a presentation on the screen makes for a nice change to reading all about it in your book.
You didn't hear this from me, but… you don't need to be using this textbook to be able to view these presentations. They're on the web for all to see, so even if you're using Maths Quest or one of the other textbooks, you can still access these activities for free!
MathsOnline
MathsOnline is a sort of online classroom for the Australian maths curriculum. It was developed by maths teachers, and best of all it's completely free. There are video lessons which are really clear with great explanations.
Not all of it relates to the VCE further maths topics, but there are a number of videos that do. I've run through a list of the ones that apply to the core (data analysis) section in a previous post to help you get started.
This books explains calculator steps using the TI-83, so they've also published calculator instructions for both the TI-Nspire CAS and the Casio Classpad CAS which you can download and print, with clear instructions and screenshots.
Merspi
Merspi is a free question and answer site for the VCE community. You post a question about further maths, or any of your other VCE subjects for that matter, and other students/teachers/tutors will answer them for you. Simple! And free, which we love around here.
Google Books
There are a few further maths textbooks with partial previews available on google books – scanned copies of the books available legally online for you to read and enjoy. The whole book isn't available in the preview, but the core section is in each of these, so you can get some more revision notes without spending a dime. The three I found were the MathsWorld textbook, the Excel revise in a month study guide and Cambridge's further maths checkpoints (2009 edition).
Patrick's Maths Tutorials
Here's another maths tutor putting free videos on the web for everyone to learn from. Most of the topics he covers are part of the Maths Methods and Specialist courses, but there are few in there that cover topics from the further maths material, and his explanations are fantastic!
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Panageos: This is a powerful Plane Analytic Geometry Problem Solver and Visualizer.
Panageos is oriented to the intensive solution of problems on Plane Analytic Geometry The main feature of Panageos is its power to read the user's equations and interpret them, for this reason the data input is exclusively through the keyboard (coefficients of several types of equations are entered via keyboard).
In order to solve problems with Panageos, both the mouse and keyboard are used, clicking the icon of an object (point, line, circle, etc) opens a small menu with several options for user's data entry for each object. Both, coordinates and coefficients of equations can be input.
Panageos reads and interprets several types of equations for the straight line, the circle, the parabola, the ellipse, and the Hyperbola.
Panageos may be used by the teacher to prepare classes, homeworks and exams. The students may use Panageos to solve hundreds of problems in a very short time, also may be used to verify solutions previously obtained with pencil on Teacher Screensaver 2.2 - Are
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$19.99Math Detective Level A Book 1
Higher-Order Thinking Reading Writing In Mathematics
By: Terri Husted
Math Detective uses topics and skills drawn from national math standards to prepare your students for advanced math courses and assessments that measure reasoning, reading comprehension, and writing in math. Students read a short... more
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Costs
Course Cost:
$175.00
Materials Cost:
None
Total Cost:
$175
Special Notes
State Course Code
02053Pre-AlgebraCOURSE DESCRIPTION:
The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics in a year long algebra course. Topics included are real numbers, simplifying real number expressions with and without variables, solving linear equations and inequalities, solving quadratic equations, graphing linear and quadratic equations, polynomials, factoring, linear patterns, linear systems of equality and inequality, simple matrices, sequences, and radicals. Assessments within the course include multiple-choice, short answer,or increase retention and expand opportunities for assessment. With each topic, diagnostic quizzes are presented to the student, allowing students to pass through areas of content. Audio readings are included with every portion of content, allowingCOURSE OBJECTIVES:
After completing this course, students will be able to:
• Read, write, evaluate, and understand the properties of mathematical expressions including real numbers, radicals, and polynomials
• Add, subtract, multiply, and divide radical expressions, polynomials, and polynomial expressions
• Read, write, solve, and graph linear and quadratic equations and inequalities
• Students will solve absolute value equations and inequalities
• Work effectively with ratios and direct and inverse variation
• Solve systems of linear equations and inequalities
• Work with arithmetic sequences and linear patterns
• Understand basic statistics including measures od central tendencies and box plots
• Understand different types of graphs, including histograms, line graphs, circle graphs, and stem-and-leaf plots
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Equations
Linear equations are equations involving only one variable, like x, and nothing complicated like powers or square roots. With this special educational program you learn how to resolve them. You can choose out of four different exercises, and to challenge your knowledge you can play the falling blocks style game 'Valgebra': By manoeuvring the falling x-terms and numbers, you resolve an equation. But take care.., if you make a mistake, you are...
EasyConicSections gives students the ability to plot the equations of lines, circles, parabolas, hyperbolas, cubics and ellipses quickly. EasyConicSections gives students the ability to plot the equations of lines, circles, parabolas, hyperbolas, cubics and ellipses quickly. They can then study the changes in the equations as they vary the parameters of the functions. Solutions to...
Transmission line equations simulation, The simulation software TL (Transmission Line) is based on solutions of Maxwell s equations in one dimension, the so-called transmission line equations. The algorithm was developed originally by Zvonko Fazarinc with later additions by Ernesto Mart n, and has been proven to be stable and correct for all possible border conditions and parameter settings.The software TL can be applied primarily to visualizeJEME is a simple and easy-to-use software designed for quickly solving scientific problems. JEME is a simple and easy-to-use software designed for quickly solving scientific problems. Subjects include Physics, Engineering, and Geometry. Future updates will add more equations and subjects, so keep checking back for updates.JEME Features:...
Edit your LaTex equations with this application. Edit your LaTex equations with this application. LaTex Equation Editor is a LaTeX equation editor for Windows with OLE Server abilitiesUsing:
1. You may use LaTeX EE as a standalone software by simply executing latexeqedit.
2. You can generate...
This software lets you create and view many different functions and equations. This software lets you create and view many different functions and equations. You can even compare them against previous equations to see how the graph is alteredBalancing Chemical Equationsenable you to verify if the chemical equations you are working on are balanced. Balancing Chemical Equationsenable you to verify if the chemical equations you are working on are balanced. Useful for both students and teacher.sSample Learning Goals
1. Balance a chemical equation.
2. Recognize that atoms are conserved in a...
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In these activities, you explore the steps involved in solving systems of linear equations. You'll make observations about the effects of those operations on the solution sets of the systems. In Part ...TI InterActive! is a new product that enables high school and college teachers and students to easily investigate ideas in mathematics and science. The purpose of this workbook is to introduce algebra... More: lessons, discussions, ratings, reviews,...
This workbook provides high school students with activities from algebra to calculus that use Texas Instruments software TI InterActive! TI InterActive! is software for the PC that combines a word pro... More: lessons, discussions, ratings, reviews,...
TI InterActive! is an integrated learning environment in which you can create interactive math and science documents. Documents may include formatted text, graphics, movies, and live integrated mathem... More: lessons, discussions, ratings, reviews,...
All the familiar capabilities of current TI scientific calculators plus a host of powerful enhancements. Designed with unique features that allow you to enter more than one calculation, compare result... More: lessons, discussions, ratings, reviews,...
TI-Nspire™ and TI-Nspire™ CAS handhelds and computer software provide students the option to use any of these as a stand-alone learning tool, at school and at home, extending the learning ... More: lessons, discussions, ratings, reviews,...
Use your TI-Nspire to consider this scenario and question: Sam and Teri have bank accounts. Sam always withdraws money; Teri always saves it. When will they have equal balances -- or will they ever?
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The PITA Principle: How to Work With (and Avoid Becoming) a Pain in the Ass
Through entertaining scenarios and real-life situations, The PITA Principle describes the different kinds of PITAs (Pain in the Ass) at work and how to cope with each. Readers are provided with a positive scenario for each type of PITA, showcasing techniques for working with this personality type. Readers then engage in a self-evaluation process, identifying their own PITA tendencies. Finally, the authors identify ways to improve upon various self-identified PITA characteristics through a cognitive-behavioral approach to change.
Give students that extra boost they need to acquire important concepts in specific areas of math. The goal of these "how to" books is to provide the information and practice necessary to master the ...
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The Matrix Algebra Tutor: Learning by Example DVD Series teaches students about matrices and explains why they're useful in mathematics. This episode teaches students Cramer's Rule in matrix algebra. An alternative method used to solve a system of equations, Cramer's Rule is especially useful because it can be applied to a wide range of problem types. Grades 9-College. 30 minutes on DVD.
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Channels: Chinese
Mathematica covers many application areas, making it perfect for use in a variety of different classes. In this screencast, you'll get an introduction to Mathematica and learn how it can help you tackle any type of problem—numeric or symbolic, theoretical or experimental, large-scale or small. Includes Chinese audio.
Creating interactive models in Mathematica allows students to explore hard-to-understand concepts, test theories, and quickly gain a deeper understanding of the materials being taught firsthand. This screencast shows you how get started creating interactive models in Mathematica. Includes Chinese audio.
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Overview - LIFE SKILLS MATH SKILL TRACK SOFTWARE SITE LICENSE
Life Skills Math makes math relevant for students in transition from school to independent living. This practical program provides comprehensive instruction that students and adults need for being self-sufficient. The full-color text focuses on using math skills in real-life situations for those who have basic computational skills but need practice in applying these skills.
Teacher's Resource Library on CD-ROM contains the Student Workbook offering dozens of reinforcement activities (also available in print), Self-Study Guide for students who want to work at their own pace, two forms of chapter tests, plus midterm and final tests. Just select and print out the materials as needed. Everything is reproducible. For Windows and Macintosh.
Teaching Strategies in Math Transparencies stimulate learning and discussion in the classroom. Graphic organizers present concepts in a meaningful, visual way and help you teach students how to manage information. Comes with instruction book and blackline masters.
With Skill Track Software, a CD-ROM program, students can review each lesson and/or chapter within the textbook at their own pace. Includes hundreds of multiple-choice items that directly relate to the textbook's content. Built-in teacher management software allows the instructor to track student progress and print reports. For Windows and Macintosh.
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Case studies
Research mathematicians work in a wide range of areas and carry out many different tasks. Common employers include private or government research laboratories, commercial manufacturing companies and universities.
The work often involves: proving deep and abstract theorems; developing mathematical descriptions (mathematical models) to explain or predict real phenomena such as the spread of cancer or the flow of liquids; and applying mathematical principles to identify trends in data sets. Applied research can also contribute to the development of a commercial product or develop intelligence about business trends.
Collaboration with other scientists and people in other commercial functions in industry is very common because the application of mathematics is so varied. Research is undertaken into a diverse range of pure and applied maths including algebra, analysis, combinatorics, differential equations, dynamic systems, geometry and topology, fluid mechanics, mathematical biology and numerical analysis.
Typical work activities
Mathematicians in commercial organisations are involved in developing new products and providing insight into business performance. They are likely to be allocated specific projects. In smaller organisations, they may be involved in all stages of the product - from concept to customer.
In academic and research organisations, projects are undertaken to develop the understanding of particular areas of maths. There are very few pure research posts in universities, and most mathematicians working in research will also have teaching responsibilities. This may involve giving lectures to large groups of students, giving tutorials to small groups and setting and marking work including examinations.
Work in both settings usually involves some office-based activities and the use of specialist computer systems.
Tasks will vary depending on the specific work environment and organisation but may include the following:
identifying solutions by learning and applying new methods, e.g. designing mathematical models that interpret data in a meaningful way;
keeping up to date with new mathematical developments and producing original mathematics research;
using specialist mathematical software such as Mathematica, Matlab or Mathcad or using software languages such as C/C++ or Visual Basic to develop programs to perform mathematical functions;
presenting findings at group and departmental meetings as well as to senior management;
attending and sometimes presenting at national and international scientific conferences and meetings in a particular field of interest;
meeting with clients throughout projects to discuss ideas and results;
advising clients on how to benefit from mathematical analysis, making recommendations based on these analyses;
writing applications for funding;
managing a research team (or group of research students in academic settings);
producing tailored solutions to business problems using innovative and existing methods as well as suggesting new ways to analyse data;
providing more sophisticated insights into available data;
sharing the implications of new research by producing regular reports on the development of work as well as writing original papers for publication in peer-reviewed scientific journals
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Holt Mathematics Course 2. he will use this skill in every math course he or she takes from this point forward. ... 10k. 6 2k. 21. 3. 5x. 6x 8. Answers: 1.
Copyright by Holt, Rinehart and Winston All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic ...
mswooalgebra2.webs.com/../HoltAlgebra 2 Practice B.pdf
Variables and Expressions (for HoltAlgebra 1, Lesson 1-1) A variable is a letter that represents a value that can ... (for HoltAlgebra 1, Lesson 1-6) The ...
Copyright by Holt, Rinehart and Winston. 101 Holt Mathematics All rights reserved. Dear Family, In this section, your child will be learning what it means for a number to be ...
go.hrw.com/../c2ch12fia.pdf
School Year _____ Catholic Schools Office Diocese of Tucson Math Curriculum Grades K-8 Based on the National Council of Teachers of Math Students And The Arizona State ...
curriculum.pdf
1 Notes to the Teacher Objectives Correlation Presents the North Carolina objectives for Algebra 2 and lists the items related to each objective that appear in the Pre-Course ...
holtmcdougal.hmhco.com/../NC_Alg2_TPP_TE.pdf
If a performance indicator is noted the topic could be assessed in current year Unless otherwise noted, the source is the Holt text Justify the reasonableness of answers ...
wits.williamsvillek12.org/../Grade-7-Holt-Alignment.pdf
Algebra 2 Text: Algebra 1 Holt, Rinehart and Winston 1 TAMISCAL HIGH SCHOOL Equipment Needed: Scientific Calculator NOTES TO THE STUDENT: In all assignments, you will need to ...
web.mac.com/../alg2hrw.pdf
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In it's purest form it can provide endless riddles and puzzles to solve and provide solutions and answers to some of life's biggest questions. And in practical ways it can help us make the best possib...
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Most of us have experienced the amusement (and possible embarrassment) that goes with standing in front of a distorted funhouse mirror. What many people don't realize is that convex and concave mirror...
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Using exciting live-action demonstrations and easy-to-understand animation, this video delves into the fundamental concepts of reflection and its relationship to light, vision, and the physical world....
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Anyone standing in front of a mirror will instantly recognize the concept of reflection at work, but to observe the process of refraction and to develop an in-depth knowledge of it is quite a differen...
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The mathematical skills children bring with them to elementary school predict both their mathematical and literacy achievement for years to come. In this video, experts from Erikson Institute's Early...
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Instructors who are looking for a way to integrate handheld technology and visual media into their algebra classes will benefit greatly from this ten-part series. In each program, internationally accl...
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What do mathematicians mean when they say that an event has a 50 percent probability of occurring? How does the study of statistics apply algebraic principles to real-world events and conditions in a...
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Most people have a basic understanding that exponential growth means rapid growth. But framing this concept in algebraic terms and applying it to concrete problems in real-world situations is a differ...
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Environmentalists, meteorologists, economists, and people in many other disciplines have always been interested in the dynamics of variables, or quantities that change-for example, the number of polar...
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Minor steps like reversing the "less than" or "greater than" sign might look simple, but when students first try to grasp the strange world of inequalities, they often feel overwhelmed. That's a probl...
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The Math Resource Centre is a free drop-in service for students who need assistance
in Mathematics. The Math Resource Centre services are available for any student who wishes
to improve their mathematical skills. The Math Resource Centre is primarily directed to those
in first year Mathematics courses, but students in any MtA course who want help in math-related
topics are welcome to use the Math Resource Centre.
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Analyzes, evaluates and responds to ethical issues from an informed personal value system.
4. Cultural Skills:
Understands culture as an evolving set of world views with diverse historical roots that provides a framework for guiding, expressing, and interpreting human behavior.
Demonstrates knowledge of the signs and symbols of another culture.
Participates in activity that broadens the student's customary way of thinking.
5. Aesthetic Skills:
Develops an aesthetic sensitivity
Further Course Goals:
From the perspective of mathematical content, this course should allow the student to expand and apply skills and knowledge gained in the first two semesters of Calculus to the topics of infinite sequences and series and to three-dimensional space.
The student will gain knowledge and skills, and the ability to apply these, to a variety of situations that might be encountered in the world of mathematics, science, or engineering.
The student will further improve his/her ability to communicate mathematical ideas and solutions to problems.
The student will improve her/his problem-solving ability.
From a most general perspective, the student should see growth in his/her mathematical maturity. The three-semester sequence of calculus courses form the foundation of any serious study of mathematics or other mathematically-oriented disciplines and this course is the capstone of that sequence.
COURSE OBJECTIVES
1. Thinking Skills:
a) Explores infinite sequences and series, and the concept of convergence and divergence.
b) Investigates applications of series.
c) Explores functions of more than one variable.
d) Develops an ability to construct 3-dimensional graphs of these functions, especially with an appropriate use of technology.
e) Learns the meaning and applications of partial differentiation.
f) Explores the use of multiple integration to find volumes and surface areas of 3-dimensional surfaces.
2. Communication Skills:
a) Collects a portfolio of one's work during the course and write a reflection paper.
b) Does group work (labs and practice exams) throughout the course, involving both written and oral communication.
c) Uses technology - graphing calculators and Maple V in the computer lab - to solve problems and to be able to communication solutions and explore options.
a) Develops an appreciation for the intellectual honesty of deductive reasoning.
b) Understands the need to do one's own work, to honestly challenge oneself to master the material.
4. Cultural Skills:
a) Develops and appreciation of the history of calculus and the role it has played in mathematics and in other disciplines.
b) Learns to use the symbolic notation correctly and appropriately.
c) Explores how people living in a pre-calculator/computer culture used convergent infinite series and sequences to evaluate expressions we take for granted because we get them at the push of a button (e.g., e, π, ln(2), ...).
5. Aesthetic Skills:
a) Develops an appreciation for the austere intellectual beauty of deductive reasoning.
b) Develops an appreciation for mathematical elegance.
Course Outline:
The student will study infinite sequences and series, exploring specifically the questions of divergence and convergence, and studying the use of such series to find the value of certain functions and certain numbers.
The student will study three-dimensional space and vectors, lines, planes and general surfaces.
The student will study derivatives and partial derivatives of functions in 3D space.
The student will study the concept and process of multiple integration, and explore a variety of applications of such processes.
COURSE PHILOSOPHY AND PROCEDURES:
I am a firm believer in the notion that you as the student must be actively engaged in the learning process and that this is best accomplished by your DOING mathematics. My job here is not to somehow convince you that I know how to do the mathematics included in the course, but to help guide YOU through the material; I am to be a "guide on the side, NOT a sage on the stage". I may not always say things in the way which best leads to your full comprehension, but I will try to do so and I ask that you try to see your learning as something YOU DO rather than as something I do TO you. PLEASE feel free to come see me to discuss questions or concerns you may have during the course; I will not bite! PLEASE ask questions in class if you have them; the only dumb question is the one you fail to ask!
Let me therefore urge you to make it a regular part of your day to try working the homework problems. There will never be enough time for us to go through every listed problem in class, and it is unrealistic to think that you will be able to find the time to work through every listed problem, but you should at least spend some time thinking about virtually every problem, and working the more interesting or challenging to completion. The daily Homework assignments will not generally be collected or graded. They are intended to structure your learning so that you regularly challenge yourself to see that you understand the material we are looking at. The important thing is that you at least look at all the assigned problems. You should also feel free to work other problems if you deem it necessary for your comprehension of the content.
In general, I think students can benefit greatly by working together on problems. While there is some danger of the "blind leading the blind" syndrome, or of students deceiving themselves into thinking they understand the material better than they actually do, for the most part grappling with ideas and trying to explain them to another student or learning to listen to others explain an idea is a wonderful way to see that you really do understand the material. In class we will spend an occasional day working on a group "lab", and we sill also typically have a group "practice exam" before the individual exams, and I also encourage you to find a "learning group" outside of class.
I will be asking you to keep a JOURNAL and a PORTFOLIO. Every other Monday (Sep 10, Sep 24, Oct 8, Oct 22, Nov 5, Nov 19, and Dec 3) morning I will collect a one-page JOURNAL entry; you should write about the concepts we are encountering and about your efforts, successes and failures in the learning process. These are intended to be personal reflections, and the 5 points for each entry is meant to demonstrate that I am placing value on this reflective writing; as long as you put obvious adequate effort into your journaling, you will get the 5 points. I would like you to type up your journal entries and leave the files in your Blackboard "Drop Box".
The PORTFOLIO will be collected at the end of the semester, on Friday 15 December. This "portfolio" should be a representative collection of your work during the semester, and should include 5 problems, written up in a finished form, along with a brief discussion of why you chose to include that particular problem. These problems should represent work you are proud of, or problems that brought you to a breakthrough point. This portfolio of your work will be worth 40 points, 8 points per problem. Presumably you will hand in 5 sheets of paper, each of which will include a nicely organized solution to the problem, which might be from homework, or from a lab or worksheet, of from an exam, along with that reflection mentioned above.
I use a rather traditional GRADING SCALE: A = 90% (or better), AB = 87%, B = 80%, BC = 77%, C = 70%, CD = 67%, D = 60%. I do try to come up with a mix of points so that exams are not weighted too heavily; about half the points during the semester will come from individual, in-class exams - the rest will come from group labs, take-home problems, group practice exams, journals, and portfolios.
It is rarely much of a problem at the level of a calculus course, but it remains important that students turn in work in a timely manner, so that they do not get behind. Consequently, LATE ASSIGNMENTS will be penalized 20% of the possible points for each class period late, up to a maximum of three periods.
AMERICANS WITH DISABILITY ACT:
If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me or Wayne Wojciechowski in MC 320 (796-3085) within 10 days to discuss your accommodation needs.
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Visitors
Why Mathematics at Guilford?
While many people associate mathematics with calculations and arithmetic, there is much more to math than simply crunching numbers. In its most general form, math is sometimes described as the science of patterns. Some of the patterns mathematicians explore include algorithms, sets, sequences, graphs, networks, functional relations, statistical data, and geometric and topological structures. Since the analysis and understanding of patterns is important in virtually every discipline, the ideas and methods of mathematics can be applied in almost any field. Sometimes mathematical analysis allows for the prediction of certain patterns (or at least of their likelihood). Other times, just as importantly, mathematics reveals that making a prediction with reasonable certainty is impossible.
Students who are well versed in math will be better prepared for employment and for graduate work in any field that deals with data analysis, quantitative reasoning, or logical deduction. Mathematics students will also be better able to understand recent advances in subjects where mathematical methods are routinely applied. Even in fields such as law and philosophy, where computational issues may not be emphasized, the use of logical thinking as required by mathematical proofs is a valuable skill.
Many majors at Guilford, including Business Management, Biology, Chemistry, and Physics, already require mathematics courses. However, the increasing use of mathematical methods and terminology in many fields, scientific and otherwise, is a great reason to study more than just the bare minimum of mathematics. Questions about infinity, higher dimensions, the limitations of computing, and the prediction of future events are just some of the topics up for grabs.
If you are a current or prospective student who wants to know more about the different math courses Guilford has to offer, please contact any of the mathematics faculty, and we'd be happy to tell you more.
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Course Number and Title
Number of Credits
Minimum Number of Instructional Minutes Per Semester
Prerequisites
Math Placement Test score of 7 or MATH103 (C or better)
Corequisites
None
Other Pertinent Information
A comprehensive departmental final examination is required for this course.
Catalog Course Description
This course provides a preparation in mathematics for students interested in elementary education. Topics include elementary logic, sets, relations, functions, numeration systems, whole numbers, integers, and number theory.
Required Course Content and Direction
Learning Goals:
define the whole numbers using equivalence classes, define the operations for whole numbers using sets, order whole numbers using sets, calculate multiples and powers of whole numbers, use mental math and estimation;
work with numeration systems including Hindu-Arabic, Roman numerals, and other ancient systems;
work with positional numeration systems; perform calculations involving positional numeration systems and other bases; and develop an understanding of the importance of place value and groupings in decimal system and other base systems;
complete prime factorizations and use this to calculate the greatest common divisor and greatest common multiple of two or more numbers; use divisibility rules to test for divisibility; and
define the integers from the whole numbers and develop an understanding of the properties of the integers using the definition of the integers.; define the operations for the integers using equivalence classes; order the integers and simplify expressions involving absolute value and negative exponents.
use methods, concepts and theories in new situations(Application Skills).
demonstrate an understanding of solving problems by:
recognizing the problem
reviewing information about the problem
developing plausible solutions
evaluating the results
Planned Sequence of Topics and/or Learning Activities:
elementary logic
inductive reasoning
deductive reasoning
patterns and pattern recognition
algorithms
sets, relations, and functions
set operations
artitions as equivalence relations and equivalence classes
unctions
whole numbers
definition of whole numbers
properties of whole numbers
operations for whole numbers using sets
ordering whole numbers using sets
multiples and powers of whole numbers
mental math and estimation
numeration systems
Hindu-Arabic, Roman numerals and other ancient systems
positional numeration systems and other bases
operations using the decimal system and other bases
number theory
factors, factorizations, and prime numbers
divisibility rules
greatest common divisor and Euclidean Algorithm
least common multiple
integers
define integers from whole numbers
properties of integers
operations for integers and their algorithms
ordering integers
absolute value
negative exponents
Assessment Methods for Core Learning Goals:
Course
Students will apply mathematical concepts and principles to identify and solve problems through informal assessment (oral communication among students and between teacher and students) and formal assessment (may include homework, quizzes, exams, projects, and comprehensive final).
Core (if applicable)
Math or Science: Assigned problems require the student to translate a descriptive problem into mathematical statement and solve.
Critical Thinking and Problem Solving: Critical thinking and problem solving skills are required when creating Venn Diagrams and evaluating the information in the diagram to answer questions posed about the problem. They are also assessed when solving Logic problems.
Other Evaluative Tools: Exams, quizzes, class participation, and projects as specified in the individual instructor's course format are utilized.
Reference, Resource, or Learning Materials to be used by Students:
A departmentally selected textbook will be used. Details will be provided by the instructor of each course section. See course format.
Teaching Methods Employed
Section VIII is not being used in new and revised syllabi as of 12/10/08.
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New Textbook
Related Products
Error Patterns in Computation: Using Error Patterns to Improve Instruction
Summary
As your students learn about mathematical operations and methods of computation, they may adopt erroneous procedures and misconceptions, despite your best efforts. This engaging book was written to model how you, the teacher, can make thoughtful analyses of your student's work, and in doing so, discover patterns in the errors they make. The text considers reasons why students may have learned erroneous procedures and presents strategies for helping those students. You will come away from the reading with a clear vision of how you can use student error patterns to gain more specific knowledge of their strengths on which to base your future instruction. Book jacket.
Table of Contents
PART ONE DIAGNOSIS AND INSTRUCTION
1
(90)
CHAPTER One Computing with Paper and Pencil in an Age of Calculators and Computers
3
(6)
Paper-and-Pencil Computation Procedures Today
3
(3)
Importance of Conceptual Understanding
6
(1)
Error Patterns in Computation
7
(2)
CHAPTER Two Diagnosing Error Patterns in Computation
9
(35)
Learning Error Patterns
10
(5)
Overgeneralizing and Overspecializing
15
(2)
Encouraging Self-Assessment
17
(3)
Using Graphic Organizers for Diagnosis
20
(2)
Using Tests as a Part of Diagnosis
22
(5)
Using Computers for Diagnosis
27
(1)
Interviewing: Observing, Recording, and Reflecting
27
(13)
Getting at a Student's Thinking
29
(3)
Observing Student Behavior
32
(1)
Recording Student Behavior
33
(1)
Watching Language: Yours and Theirs
34
(1)
Probing for Key Understandings
35
(3)
Designing Questions and Tasks
38
(2)
Guiding Diagnosis of Written Computation
40
(4)
CHAPTER Three Providing Needed Instruction in Computation
44
(49)
Understanding Concepts and Principles
46
(4)
Numeration
47
(1)
Equals
48
(1)
Other Concepts and Principles
49
(1)
Acquiring Specialized Vocabulary
50
(1)
Using Models and Manipulatives
51
(3)
Recalling the Basic Facts
54
(4)
Stressing Estimation
58
(2)
Teaching Students to Compute
60
(6)
Talking and Writing Mathematics
66
(3)
Using Graphic Organizers for Instruction
69
(5)
Using Calculators and Computers
74
(2)
Using Alternative Algorithms
76
(2)
Subtraction of Whole Numbers: The Equal Additions Method
77
(1)
Subtraction of Rational Numbers: The Equal Additions Method
77
(1)
Using Cooperative Groups
78
(2)
Monitoring Progress with Portfolios
80
(2)
Guiding Instruction
82
(9)
Focus on the Student
82
(1)
Teach Concept and Skills
83
(1)
Provide Instruction
83
(1)
Use Concrete Materials
84
(1)
Provide Practice
85
(6)
PART TWO IDENTIFYING, ANALYZING, AND HELPING STUDENTS CORRECT SPECIFIC ERROR PATTERNS
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Batten is great for the Life Con and Survival Models stuff if you already have some knowledge of the material. He just provides a lot of problems and some useful shortcuts that are not taught in the Bowers text. So if you have no prior knowledge of the material than I would not use Batten.
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