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Introduction to Algebraic Geometry | UCLA Extension in Fall 2017
Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert's Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.
Yes math fans, as previously hinted at in prior conversations, we'll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won't happen until July 31, 2017.
While it's not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we'll post them here first. As of this writing, there is no officially announced textbook for the course, but we've got some initial guesses and the best are as follows (roughly in decreasing order):
Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he's that good. This class, my 22nd, will be the start of my second decade of math with him.) | 677.169 | 1 |
When dealing with courses that involve complex systems and formulas, it is essential to find teaching tools that easily handle detailed and intricate calculations, while simplifying the process to minimize time and effort for the students. | 677.169 | 1 |
Rediscovering Mathematics: You Do the Math
By Shai Simonson
Rediscovering Mathematics is a guide to more effective mathematical education, offering topics and puzzles that can inspire a renewed interest in mathematics among students and enthusiasts.
Suitable for anyone who wants to expand their view of mathematics, the book's topics are linked by the theme of mathematical learning through investigation, challenge, and discovery. It can be used as a text for training mathematics teachers at all levels, and is recommended for anyone interested in mathematics and how mathematics should be taught.
Containing dozens of illustrations, tables, and exercises, the book focuses on problem solving and understanding, subsuming learning by memorization. Topics include number theory and applications to secure Internet communication and to probability in sports and gambling.
Special Features:
Nearly 200 end-of-chapter exercises
Numerous challenges with hints and solutions
The focus on learning through problem-solving and understanding is enhanced by linking mathematical topics to everyday life
Table of Contents
Acknowledgements
A Guide for the Reader
Introduction - how to read mathematics
1. Mathematical discovery in the classroom
2. Don't reach for your calculator (yet)
3. Have another piece of pie, Zeno?
4. Thinking like a mathematician - lessons from a medieval Rabbi
5. What is mathematics good for?
6. Three averages
7. Algorithms - the unexpected role of pure mathematics
8. Pythagoras' Theorem and math by pictures
9. Memorizing versus understanding
10. Games and gambling
11. Soccer balls and counting tricks
12. Pizza Pi and area
13. Back to the classroom
Resources for rediscovering mathematics
Further reading
Index
Excerpt: Memorizing Versus Understanding (p. 129)
Memorizing mathematics without comprehension is often harmful. If you memorize a poem that you don't understand, there is still the chance that the flow of the words may have an effect on you. When you memorize dates of historical events, at least you know the chronological order of those events, even if you may not know their significance. When you memorize mathematics without understanding, you delude yourself into thinking that you know something, when in fact you do not. This delusion compounds the lack of understanding. Your ability to apply the knowledge, generalize it, or even question its truth is compromised.
About the Author
Shai Simonson (Stonehill College), who has lectured all over the world, has published articles on mathematics education, the history of mathematics, popular mathematics, theoretical computer science, and computer science education. | 677.169 | 1 |
8 1 Study Guide And Intervention Answers 134084
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An Introduction to Gaussian Geometry
Description: The purpose of these notes is to introduce the beautiful theory of Gaussian geometry i.e. the theory of curves and surfaces in three dimensional Euclidean space. The text is written for students with a good understanding of linear algebra, real analysis of several variables, and basic knowledge of the classical theory of ordinary differential equations and some topology.
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Tight and Taut Submanifolds by Thomas E. Cecil, Shiing-shen Chern - Cambridge University Press Tight and taut submanifolds form an important class of manifolds with special curvature properties, one that has been studied intensively by differential geometers since the 1950's. This book contains six articles by leading experts in the field. (6058 views)
Synthetic Differential Geometry by Anders Kock - Cambridge University Press Synthetic differential geometry is a method of reasoning in differential geometry and calculus. This book is the second edition of Anders Kock's classical text, many notes have been included commenting on new developments. (7593 views) | 677.169 | 1 |
SLO 1. Upon completion of the
course students will be able to apply basic algebraic concepts of relations
and functions.
-Students will state the distance using the distance formula.
-Students will state the midpoint using the midpoint formula.
-Students will state information from the graph of a function.
-Students will Graph eight basic functions using transformation.
-Students will state if a function is one-to-one.
-Students will state the inverse function.
-Students will graph inverse functions.
-Students will recognize the equation of a circle.
-Students will graph circles.
SLO 2. Upon completion of the course students will be able to evaluate
trigonometric functions.
-Students will change degrees to radians.
-Students will change radians to degrees.
-Students will determine arc length.
-Students will determine the six trig functions from a unit circle.
-Students will determine trig values for 30*, 45* and 60* angles.
-Students will determine trig values for quadrantal angles.
-Students will determine the six trig functions from a circle with radius r.
SLO 3. Upon completion of the course students will be able to apply and
identify the properties of trigonometric functions.
-Students will determine the domain and range of the trig functions.
-Students will determine the fundamental period of the trig functions.
-Students will find values of the trig functions using fundamental
identities.
-Students will find values of the trig functions using even and odd
properties.
SLO 4. Upon completion of the course students will be able to graph
trigonometric functions.
-Students will determine information from the graph of a function.
-Students will graph trig functions.
-Students will determine the amplitude and period of a sinusoidal function.
-Students will graph sinusoidal functions using key points.
-Students will apply transformations to graphs of trig functions.
-Students will determine the phase shift of a sinusoidal function.
SLO 5. Upon completion of the course students will be able to apply analytic
trigonometric concepts.
-Students will find values of inverse trig functions.
-Students will establish identities using fundamental identities.
-Students will find values using sum and difference formulas.
-Students will find values using double-angle and half-angle formulas.
-Students will find values using product-to-sum and sum-to-product formulas.
-Students will solve trigonometric equations.
SLO 6. Upon completion of the course students will be able to solve
applications of trigonometric functions.
-Students will find values of trig functions using right triangles.
-Students will determine the bearing of an object.
-Students will use the law of sines to solve triangles.
-Students will use the law of cosines to solve triangles.
-Students will find the area of a triangle using trig functions.
-Students will find the area of a triangle using Heron's formula.
-Students will find a unit vector.
-Students will find a vector from its direction and magnitude.
-Students will find the dot product two vectors.
-Students will find the angle between two vectors.
-Students will decompose a vector into two orthogonal vectors.
SLO 7. Upon completion of the course students will be able to apply the
properties of exponential and logarithmic functions.
-Students will convert equations between exponential and logarithmic form.
-Students will evaluate common and natural logarithms.
-Students will identify and graph exponential and logarithmic functions.
-Students will graph common and natural logarithmic functions.
-Students will apply the properties of logarithms to evaluate expressions.
SLO 7. Upon completion of the course students will be able to solve
exponential and logarithmic functions.
-Students will solve exponential functions.
-Students will solve logarithmic functions.
Evaluation/Assessment
Practices
This course will be
taught primarily by lecture and demonstration. The textbook will be the basis
for material to be presented and the student will be responsible for reading
all material and working all problems assigned by the instructor. Computer
assisted instruction disks and videos which are located in the Learning
Resource Center (LRC), is available for supplementary purposes.
Assignments and Course Format:
1.1 Multiples of 3
from 12 through 42
1.2 Multiples of 3
from12 through 84
1.3 Multiples of 3
from18 through 69
1.4 Multiples of 3
from 9 through 54
1.5 Multiples of 3
from 9 through 36
1.6 Multiples of 3
from 9 through 72
1.7 Multiples of 3
from 6 through 57
2.1 Multiples of 3
from 12 through 72
2.2 Multiples of 3
from 12 through 81
2.3 Multiples of 3
from 12 through 87
2.4 Multiples of 3
from 9 through 72
2.5 Multiples of 3
from 9 through 45
2.6 Multiples of 3
through 24
3.1 Multiples of 3
through 45
3.2 Multiples of 3
through 54
3.3 Multiples of 3
through 63
3.4 Multiples of 3
through 54
3.5 Multiples of 3
through 51
3.6 Multiples of 3
through 36
3.7 Multiples of 3
through 36
3.8 Multiples of 3
through 36
4.1 Multiples of 3
through 51
4.2 Multiples of 3
through 42
4.3 Multiples of 3
through 30, and 44,45
4.4 Multiples of 3
from 6 through 24, and 39,40
5.4 Multiples of 3
through 54
5.5 Multiples of 3
through 33
7.1 Multiples of 3
through 72
7.2 Multiples of 3
through 75, and 90,93,99
7.3 Multiples of 3
through 39, and 45,58,62
7.4 Multiples of 3 through 24, and 51,52,60
Grade
Scale:
The following scale
will be used to determine a letter grade:
90% - 100% > A
80% - 89% > B
70% - 79% > C
60% - 69% > D
Below 60% > F
Grading
Policies:
At least four 50
minute objective examinations will be given during the semester. Problems are
assigned from each chapter taught and a grade is given for each assignment.
The test average will comprise 80% of the semester grade with the remaining
20% of the grade determined by the assigned problems. If five tests are given
the lowest test excluding the final will be dropped and not counted in the
final average. Three of the lowest homework scores will also be discarded and
no late homework will be accepted. If a test is missed it will be in the
instructors discretion whether that test will be made up or the test allowed
to drop.
Expectations:
The student is
expected to take all examinations at the specified time and date. If a student
has circumstances arise which make it impractical to take a test at its
regularly assigned time it is his or her responsibility to contact the
instructor prior to the examination if possible. The student will be given a
make- up examination if, in the instructors opinion, the student had a
legitimate reason to miss the regularly scheduled test. Make-up tests are
strongly discouraged.
Attendance:
Attendance is highly important and stressed to
each student. Attendance is not counted as part of a grade;
however, poor attendance usually reflects poor performance. Students who
begin to establish irregular habits are reminded of their need to attend
regularly
Additional Course Information:
Before class sessions
begin, completely turn off power on all cell phones and other electronic
devices and put them out of sight. Some allowances may be made for phones if
it is truly important for students to have quick communication (a sick child,
for example), but in such cases, students must inform the instructor before
class begins, and the phone must be switched to a manner/silent mode.
Otherwise, do not take calls, make calls, check text messages, send text
messages, listen to music, or use any other applications during class sessions.
Some exceptions may be made as long as devices are used in direct connection
to the class, taking notes or researching a class topic, for example.
Reading e-mail, sending e-mail, checking/using social networking sites, or
any other uses not directly related to a class are not permitted.
Depending on the exact nature of the offense, students who violate this
policy regarding cell phones, laptop PCs, or any other electronic devices
should expect a penalty, which may range from being dismissed to being dropped
from the class. No cell phones or any electronic devices other than
handheld calculators are allowed on tests.
SERVICES,
POLICY, and PROCEDURES:
Student
Email:
I
MPORTANT- All course information, billing,
financial aid notices, housing information, scholarship awards, degree check
results, and other mail will be sent to you via student email. Please
remember to check your student email often for important information.
ADA statement: Carl Albert
State College complies with Section 504 of the Rehabilitation Act of 1973 and
the Americans with Disabilities Act of 1990. Students with disabilities who
need special accommodations should make their request in the following
way:
* Talk
with your instructor after class about your disability or special needs
related to work in class.
Poteau Campus
* Complete the Request
for Special Accommodations Form with the Student Disability Services Coordinator. The Poteau office is located
in the Ollie Building on the third floor office number 1308. You can also
reach me in the LRC/NARC room 1318. Phone is 918-647-1319
Sallisaw Campus
* Complete
the Request for Special Accommodations Form with the Assistant Student
Disability Services Coordinator located in the Learning Resource Center in
office SC 8025. .
FOR WEB COURSES
* Call or
e-mail your instructor about your disability or special needs related to work
in web courses.
* Complete the Request for
Special Accommodations Form with the Student Disability Services Coordinator.
You may find information on our website under Student Affairs/Student
Disability Services.
Services
•LRC: The Learning Resource Center is
located in the George Ollie Center for Academic Excellence on the third
floor, in room 1301; Crissy Keeton, the LRC director, may be reached at
918-647-1319. The LRC offers tutoring in a variety of subjects including math
and English: specific tutoring schedules are available.The LRC also offers notes and textbooks for
many classes, as well as a computer lab and study area. Hours of operation
are Monday through Thursdays from 8 am to 6 p.m., and on Fridays from 8 am to
4 pm.
•Health Services: CASC Student
Affairs/Student Life is designated as the first stop for students/employees
in assisting them with primary care health services and holistic health
prevention measures through a partnership with Stigler Health & Wellness
Center, Inc. Please contact the Office of Student Affairs/Student Life for a
"Healthcare Eligibility Form".
•Student
Counseling Services: Carl Albert State College is committed to assisting
students in all areas of their lives. Student success and well-being is of
the utmost importance and it is our goal to improve the quality of life so
that personal and academic growth can take place. CASC suggests students
utilize the preventative counseling services that are provided through the
Oklahoma Department of Mental Health and Substance Abuse.
In addition the following local agency, Cavanal
Counseling, will provide free counseling services for victims of domestic
violence, dating violence, sexual assault, and stalking. Choctaw Nation has
provided a grant to cover these costs for all CASC students. For additional
assistance, please contact the Office of Student Affairs/Student Life in
Hemphill Hall at 918-647-1370.
·Library: Research for your class should be conducted at
the CASC Libraries. College-level research requires college-level
sources. CASC Libraries offer a number of appropriate sources in both print and
electronic formats. Visit the library in person for research assistance
or at
The National Postsecondary Education
Cooperative (NPEC) issued Information Required to Be Disclosed Under the
Higher Education Act of 1965: Suggestions for Dissemination (NPEC 2010-831).
This publication is available at
In
the event class must be cancelled by the instructor the student will be
notified through various methods including, but not limited, to the
following: text message, email, or written notification. Students
should check their Carl Albert email accounts regularly for such
notifications. When possible, instructors will provide notification in
advance.
In
instances of school closure the notification process occurs in the following
ways: the alert system is used to send messages including phone calls,
text messages, and emails to all names in the alert system as soon as a
decision has been made regarding the status of CASC; an email is sent to all
Carl Albert email addresses; closure information is posted to the CASC
website as quickly as possible; the phone message for incoming calls at the
CASC switchboard will indicate closed status; local radio stations and
television stations are notified; however television may or may not post our
information so please be sure to check other sources of information as listed
above.
Assessment
Statement
Assessment is the process that evaluates the learning
experience with the purpose of continual improvement and has
the objective of assuring the accomplishment of the mission of
Carl Albert State College.
Academic
Integrity/ Misconduct Policy:
The following will apply in connection with academic
dishonesty:
A. The instructor and his/her Division Chairperson have
final authority over the grades given to students or the lowering
of grades because of cheating or plagiarism.
B. The term "cheating" includes, but is not limited to:
1.The use of any unauthorized assistance in taking quizzes,
tests, or examinations.
2.Dependence upon the aid of sources beyond those authorized
by the instructor in writing papers, preparing reports, solving problems, or
carrying out other assignments.
3. Acquisition, without permission, of tests
or other academic material belonging to a member of the college faculty or
staff. The term "plagiarism" includes, but is not limited to, the use,
by paraphrase or direct quotation, of the published or unpublished
work of another person without full and clear acknowledgment. It also
includes the unacknowledged use of materials prepared by another person or
agency engaged in the selling of term papers or other academic materials.
If it is established that cheating or plagiarism has more
than likely occurred:
A. The instructor may take appropriate disciplinary
action, which may include the awarding of an "F" on the particular assignment
or in the course.
B. The instructor will make a report of the incident and
of action taken to the Vice President for Academic Affairs.
C. The student will receive a copy of the report if s/he
desires and may appeal the decision of the instructor to the Academic Affairs
Committee.
D. The student and instructor may meet individually with
the Academic Affairs Committee to present documentation pertinent to the
appeal. Once the Academic Affairs Committee renders its decision, the appeal
process is concluded.
Carl Albert State College considers all forms of academic
misconduct and dishonesty serious matters which warrant serious attention.
Academic dishonesty includes, but is not limited to, cases of cheating and
plagiarism, and is, at the very least, subject to disciplinary action by the
instructor of record. More serious infractions will warrant disciplinary
actions by the college.
Plagiarism is considered unacceptable and incompatible
with the educational mission of Carl Albert State College. Since plagiarism
always carries consequences, all students are expected to be familiar with
the rules for avoiding plagiarism.
Intentional plagiarism
is a deliberate act of academic dishonesty in which an individual knowingly
represents the work or knowledge of another person as one's own, knowingly
incorporates into one's work the words or ideas of another person without
clear attribution, fails to acknowledge clearly the partial or full
authorship of someone else when submitting a work, and/or consistently fails
to cite or quote textual resources properly.
Cheating is considered to be a serious infraction of
academic integrity and as such is not tolerated at CASC. Specifically, cheating includes, but is not limited
to, instances where work is turned in that is not one's own, copying others'
answers in exams and/or papers, infiltration of grading systems, use of
deception in acquisition of answers, and/or instances of forgery.
Grade
Protest
Students may challenge a final grade, provided a solution
cannot be reached through proper academic channels. Students should first contact
their instructor and then the Division Chair if resolution is not
satisfactory. Appeals for the purpose of challenging a final grade must be
made to the Academic Affairs Committee within 90 days after the grade in
question appears on the permanent record. Information concerning
procedures to be followed is available from the Office of Admissions and
Records. (CASC Catalog, p. 74)
Faculty
Complaints
A student who feels he or she has serious grounds and
evidence to demonstrate unfair treatment by a faculty member may file a
formal written complaint with the division chair. First, however, the student
is encouraged to visit with the faculty member on an informal basis to
discuss the situation. If, after that visit, the student still wishes to file
a formal complaint, the division chair will call a meeting between the
student and the faculty member to discuss the complaint and any further
action. If the issue still remains unresolved, the division chair, faculty
member, and student will meet with the Associate Vice President of
Instruction in the Academic Affairs Office.
Semester
Regulations Concerning the Beginning and End of the Term
Students should keep in mind that the semester
begins with the first day of class, and ends with the last day of the designated
final exam period. For that reason, and because final exams may be
scheduled up to and through the last day of the final exam schedule, students
should not plan to travel until the first day after the end of finals week,
unless approval is granted by the division chair AND the Academic Affairs
Office. Students may request changes in individual final exam times in
writing and through completion of the Change in Final Exam form. Forms are
located in the office of the registrar. Requests for change for personal
convenience are generally not approved. Approvals are normally limited for
the following reasons:
Conflict with working hours on a job that has been
held during the term, and for which working schedules cannot be readily
adjusted.
Religious reasons.
Four finals in one day. Where amicable agreement
cannot be reached by the student and the instructors, the division
chair and/or Academic Affairs Office can grant accommodations.
Military obligations verified in writing.
Other exceptional hardship cases including health
reasons concerning immediate members of the household and/or death of
an immediate family member or attendance of a funeral of an immediate
family member.
Additionally, with take-home final exams, instructors have the
option and may choose to make those due on the last day of finals.
Students are required to:
1. complete the
Change in Final Exam form;
2. request the
signature and approval of the designated instructor;
3. submit the form
to the appropriate division chair for approval;
4. The form will be
forwarded to the Academic Affairs Office or designee (Vice President for
Sallisaw campus).
5. The Academic
Affairs office will then inform the student of the results of the request.
*Requests will be finalized within 72 hours of
the formal request.
Withdrawal Policy: Students withdrawing from courses
should first consult instructors and refer to the current student handbook or
website for withdrawal procedures. Additionally, the student should
contact the offices of Financial Aid, Admissions, Business, and
Retention. Students failing to attend initial class meetings will be
dropped from the class without notification. Beyond that, failure to
attend class is not equivalent to dropping the class; students who fail to
formally drop the class will receive a grade in the course.
Online
Etiquette Statement:
Carl Albert State College expects online users to follow
the same basic rules that apply in face-to-face communication. The
following guidelines provide direction for students using Internet-based
communication. Failure to follow appropriate communication rules may
result in negative consequences.
1.Think before you write. What you say online is
permanent. Review and edit before you post and take the feeling of
others into consideration.
2.Be friendly and positive. Even if you disagree with
an idea there are ways you can approach your criticism without being hurtful.
3.Use standard English. Avoid slang and jargon with
which others in the class may be unfamiliar. Communication is only
effective if the audience can relate to it.
4.Be professional. Avoid writing in all caps, using
multiple exclamation or question marks, and emoticons.
5.Ask for help. If you feel lost, or need
clarification, ask. If you don't ask the questions your instructor and
other students won't be able to respond. Besides, you're probably not
alone, but don't wait for someone else to ask for you.
Statement
of Instructor Modification Right
This syllabus is subject to alteration at the discretion
of the instructor. Notification of alteration will be provided to
students via class announcement, e-mail, blackboard posting, or similar
reasonable method.
Student Financial Responsibility
Statement:
In addition to enrolling in
classes, part of your enrollment responsibility is payment of your Business
Office Account (tuition, fees, etc.)
If
you have already paid your entire balance for the semester, and any past
balances, thank you.
If
you have not Carl Albert State College requires all students to either pay
for their Business Office Account by the first day of class or enroll in the
Nelnet Payment Plan located on the website by the first day of class.
Students
who anticipate receiving financial aid must enroll in the Payment Plan as
well. If the financial aid pays for all costs then the payment plan
will not go into effect.
If
you have questions, please feel free to call the Business Office at 918-647-1325 | 677.169 | 1 |
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Synopsis: These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short exercises that focus on developing a particular skill, mostly requiring students to draw or interpret sketches and graphs. New to the Fourth Edition are exercises that provide guided practice for the textbook's Model boxes.
Book Description Pearson College Div 42146561
Book Description029061
Book Description Pearson, 2016. Paperback. Condition: Used: Very Good. All books ship SAME or NEXT business day!! 2nd Day Shipping Available! May show signs of use but will NOT contain highlighting or underlining! Although may NOT include supplemental materials, cds, codes etc. Contact us with any questions!!. Seller Inventory # 37358
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Linear Algebra by Paul Dawkins
Description: These topics are covered: Systems of Equations and Matrices; Determinants; Euclidean n-space; Vector Spaces; Eigenvalues and Eigenvectors. These notes do assume that the reader has a good working knowledge of basic Algebra.
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Linear Algebra for Informatics by José Figueroa-O'Farrill - The University of Edinburgh These are the lecture notes and tutorial problems for the Linear Algebra module. The text is divided into three parts: (1) real vector spaces and their linear maps; (2) univariate polynomials; (3) introduction to algebraic coding theory. (7576 views)
Linear Algebra by David Cherney, Tom Denton, Andrew Waldron - UC Davis This textbook is suitable for a sophomore level linear algebra course taught in about twenty-five lectures. It is designed both for engineering and science majors, but has enough abstraction to be useful for potential math majors. (2411 views)
Basic Linear Algebra by Andrew Baker - University of Glasgow The text covers basic ideas and techniques of Linear Algebra that are applicable in many subjects including the physical and chemical sciences, and statistics. These notes were originally written for a course at the University of Glasgow. (9663 views) | 677.169 | 1 |
Highlights of Calculus is a series of short videos that introduces the basic ideas of calculus — how it works and why it is...
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Highlights of Calculus is a series of short videos that introduces the basic ideas of calculus — how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.In addition to the videos, there are summary slides and practice problems complete with an audio narration by Professor Strang. You can find these resources to the right of each video.This resource is also available on Highlights for High School.About the InstructorProfessor Gilbert Strang is a renowned mathematics professor who has taught at MIT since 1962. Read more about Prof. StrangAcknowledgementsSpecial thanks to Professor J.C. Nave for his help and advice on the development and recording of this program.The video editing was funded by the Lord Foundation of Massachusetts.
Material Type:
Online Course
Author:
Prof. Gilbert Strang
Date Added:
Jun 09, 2011 intended to provide extra learning resources in algebra for middle school and high school students. The...
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This website is intended to provide extra learning resources in algebra for middle school and high school students. The approach is to teach math concepts in basic terms using examples and diagrams, if necessary.
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Tutorial
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Jun 19, 2012 Center for Education Statistics (NCES) provides interactive software to create five different types of graphs...
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The National Center for Education Statistics (NCES) provides interactive software to create five different types of graphs and print and/or save the results: pie, bar, line, area, and x-y graphs.
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Simulation
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Unknown
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Feb 03 Create a GraphAuthor are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that...
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״Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my "class notes", they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations.I've tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes.״
Material Type:
Open (Access) Textbook
Author:
Paul Dawkins
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Jun 10, 2011 It provides the user with an interactive graphical...
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This applet is a sub-site of the Math Applets for Calculus at SLU site. It provides the user with an interactive graphical interface through which he can create different hyperbolas and ellipsis. The shapes and types of conics are changed by manually changing crucial parameters (i.e., focal distance, axis, etc.)
Material Type:
Simulation
Author:
Mike May
Date Added:
May 01, 2007 Ellipse and Hyperbola Review to Bookmark material or Course ePortfolio | 677.169 | 1 |
34 comments:
Zaccaro does a good job of creating in depth avenues for gt students b/c he makes the learning relevant. He interests students with his introduction scenarios/stories (p.5, 105, 116, 128, 165, 210, 234, 260-261). He shows how to take what you already know (or may have just learned) and take it further, but he also explains how this math is actually relevant in the world and not just in the classroom. His intros also give tribute to many of those fabulous mathematicians who have helped us get where we are today mathematically, but also helps them think "I'm going to be the next big mathematician. I will make a new discovery." His examples and provided questions provide the extra stimulation that the gt kids need.
This book is geared for the upper elementary student as well as the middle school student. The subjects that Zaccaro introduces in this book are not typically found in the elementary classroom. For example, the chapters on Percents, Ration and Proportion, Statistics, Graphing Equations, Acceleration, and Calculus are not usually introduced until later in the math classroom. For a mathematically gifted student to have the opportunity and the joy of being able to discover and work with these concepts, is truly wonderful. When I have presented my gifted students with new, advanced challenges, they absolutely eat it up. This book helps in that effort. Zaccaro also makes the lessons interesting because he presents background information, often in the form of historical notes, and does it in a manner that kids would find interesting and motivating. His use of Albert Einstein throughout the book, not only helps students to pay extra attention to what Einstein has to say, but also makes is alright to be smart, and it's okay to be a little "nerdy." As a society, we don't think twice about recognizing athletic or artistic talent, and giving those students extra opportunities to develop those talents. I just don't see us readily doing the same for the mathematically gifted student. As teachers of gifted students, we all need to be strong advocates for our students, and looking for and providing advanced learning opportunities for them.
The basic structure of the book, which includes interesting, and sometimes historical, introductions of concepts (p.5, 53, 84, 165, 260, 293), "math bites" from Einstein and the Gumbys (p.6, 8, 74, 171, 202, 302), and a plethora of problems which promote higher order thinking (three levels of problems in each chapter and 20 math contests) provide rich, in-depth avenues for all students, including gt students. The introduction of the book states "When children first see the wonders of math and science, it is as if they stepped into a room that they didn't know existed." I think that all children should be given that opportunity.
In response to nlopez: I agree with your statement that, when problems are challenging and "real world relevant", success could engender the thought that "I'm going to be the next big mathematician. I will make a new discovery". In the words of Ed Zaccaro, "Children who have a special capacity must learn to treasure and value their gift". It would be exciting to be a guide along the road of discovery for those students.
Zaccaro creates a rich and in-depth avenue for gifted students to learn and experience mathematics by giving students interesting history on math concepts and connecting them to their everyday situations such as on pages 5, 165, 280, 293. Zaccaro is also able to take what the students already know and strengthen and deepen their knowledge through his level of questions.
In response to PKassir: I agree with you in the aspect that the concepts in this book aren't really taught in the younger elementary grades, however I feel that a majority of the concepts can be tied into 5th grade TEKS. I also think the several of the sections can be simplified to meet the younger grade levels of GT students in small groups.
nlopez in response to sluther on 7/25/10 in regards to PKassir- this book is "challenging" for elementary, especially lower elem. As a 4th grade teacher, I did find many opportunities/insights that I will be able to incorporate. Also having experience with 2nd grade, depending on the level of giftedness, I can see some being applied to 2nd and 3rd grade. Definitely easier ti incorporate with 4th on up though.
nlopez in response to tatumt on 7/24/10- My turn to quote you: "It would be exciting to be a guide along the road of discovery for those students." It's these small moments that give us the biggest reward and remind us why we have chosen to be teachers.
Reply to tatumt I'm so glad that you made the statement that Zaccaro's book can be used by children of varying levels. I love presenting kids that do not have the GT label with problems like Zaccaro's. It's a great feeling to be able to introduce challenging problems to them as well. Sometimes, they don't embrace these problems, but when they do, it's so rewarding to see the expression and enthusiasm in their eyes. Thank you for reminding us all that we can present these opportunities to children of many levels.
I think the beauty of this design is that it can be geared to gifted kids of all ages. As a teacher of 7th graders, I felt that a great deal of the material could be used in my classroom. Then I started reading the other posts to see that elementary school teachers like the book and want to use it as well. That is the epitome of a book written to all levels that can challenge students in whatever grade they happen to be labeled by age.
I agree with you that this book is a great resource for all kids and not just the GT ones. My classroom is integrated GT/Pre-AP and I don't have many opportunities to work with just the GT kiddos. However, so many of these questions would be great for all the kids in my classroom to chew on for awhile.
Edward Zaccaro creates rich and in-depth avenues for gifted students to learn and experience mathematics because he makes the content applicable to the real. By sharing the history of the concept and its' inventor, student can see why and how the concept will apply to the real world not just learning it because the teacher said to. The intro to each chapter and the stories that go along with the chapter help make what could be difficult subjects fun to learn. I really like the Einstein boxes that give more insight and reminds you of the important facts to remember.
In response to sluther... I think tat is the key to this whole book for upper elementary. It allows them to experience math they would normally not do until they get older. That is what is so great about this book. Zaccaro really goes in depth at the right level for upper elementary students to be able to really enjoy and understand the concept in this book
The book does a great job of weaving real world applications, historical significance and modeling to create depth. Some examples of this are page 5 and 53, the author uses historical facts to introduce the topic. This creates interest and excitement for the students.
Zaccaro provides rich and in depth situations for gifted students to learn math concepts by introducing concepts within stories and relevant situations, as well as have cartoons and thought bubbles throughout the text. For instance, on page 105 he ties the history behind the mathematics when discussing perimeters and circumferences to make the concepts realistic, rather than just learning a theory without the reasoning or real-life example behind it.
Zaccaro presents the math concepts in a way that students want to devour it. He hooks them with background knowledge, history, and real world application. Then he describes the steps for solving steps in a rigorous way! He then adds tips to simplify even the most complex concept. Page 108 really breaks the Pythagorean Theorem down after page 105 introduces the facts about perimeter and circumference.
In response to ratliffb on July 28th, I agree with how well the book is written. It is simple enough that anyone could read it to learn the concepts. I think the different leveled problems wouldn't be appropriate for everyone, which is why we differentiate!
Mr. Zaccaro does it logically and somewhat sequentially but he uses the fundamentals with examples and humor and then jumps the student up to application level very quickly.He also provides various levels so that students can progress on their own with any particular type of problem. On page 67 he starts with level 1 problems, then on the next page offers level 2 problems, and then moves on to the Einstein level as he does with each section of new subject matter. He also uses lots of graphics to explain/demonstrate the examples. For example the Venn diagrams on p.30 and the function machines on page 35 alongside the cartoon characters and their speech bubbles. The other thing that is done frequently is in the chapter introductions where he gives background information or examples of application (giving the big idea) before showing the fundamentals. On p.165 he does this with the trig section.
In response to kohlerj, I do agree that the beauty of this book is that it is applicable to various age groups and ability levels. There is such a wealth of problems that it can be tailored to meet class ors tudent needs.
Zaccaro creates rich and in-depth avenues for gifted students to learn and to experience mathematics because he understands their ability to understand concepts quickly and to make connections. He challenges, in a fun way, the children to connect and generalize learned information to new situations. For example: On pg. 110 the children weren't shown a strategy to make separate polygons out of the overall, original one, but they will discover this through practice and manipulation on their own; On pg. 109 (#3), a radical symbol on a calculator is needed to solve because 20 isn't a perfect square. The children will either learn to write the answer in its radical form, or they will learn to use that function on a calculator; On pg. 111 the kids have to learn that d = 2r.
The author creates rich and in-depth avenues for gifted students to learn math by making the concepts relevant and challenging. He makes the concepts interesting and fun. He introduces each concept with some history and humor and then gives them different levels to work with either independently or with a group. I like the way the author introduces the Metric System on page 53. He first explains that in the customary system the measurements are based on the human body making it difficult to use. He also explains why the metric system makes more sense and is easier to use.
I agree with Pkassir that alot of the this material is not usually presented in an elementary classroom, and the GT students would love learning something new.I also agree with the fact that many times GT students are not given extra opportunities to develop their academic skills.
I believe the way that he presents information that seems to be so confusing such as Chapt. 19 dealing with Calculus. I never experienced Calculus in high school, but it was in his explanation and connections that made this concept easier for me to understand. I think the kids are really going to be intrigued with all they are learning, as well as searching for ways to apply their knowledge to real life.
In response to PKassir, I agree that much of this content is applicable to 5th grade and I think the history connection is going to give fuel to the fire of getting the kids excited about learning more math/science together.
I think the author did a great job in creating rich and in-depth avenues for GT students. I teach 6th grade pre-ap and after studying this book I feel more confident teaching 8th grade Algebra (which is 9th grade in Academic class) concepts to my GT students. Author makes advanced concepts such as solving simultaneous equations, trigonometry etc. understandable, enjoyable and interesting for GT students. He builds the foundation by helping GT students see the big picture, introduces the concept without repetition and then let them explore much deeper by various levels of challenging questions.
In response to SharonG on July 29th, I agree that the author made calculus easier to understand. The way he present concepts and make connections are very helpful in building stamina needed to tackle advanced concepts such as calculus.
Zaccaro creates rich and in-depth avenues for gifted students in many ways throughout the book such as in chapter 8 about Area. (I picked this chapter because I teach 4th grade and it is one of the most relevant to my grade) In this chapter, Zacarro introduces in a very effective and concise way the area of many objects like rectangles, triangles, circles, and odd figures allowing the student to have a greater depth about the concept of Area. Area is an easy concept to grasp for GT students so this will be a great resource for them instead of doing excess drill and review of just rectangles and odd figures. The real life problems of page 122-127 are great ways to stimulate the brain of the GTstudent.
In response to Kohlerj on July 28, ..Then I started reading the other posts to see that elementary school teachers like the book and want to use it as well. That is the epitome of a book written to all levels that can challenge students in whatever grade they happen to be labeled by age.. Indeed, this is a great book to be used to have the gt student experience intellectual frustration in a positive way and enjoy the music of math. After taking the online GT course course with Dr. Karen Rogers about the 10 Options in GT Education-a Synthesis of Research, Im more assured of the treasure of this book because it agrees with several of her statements such as Exposing the GT student to content beyond their grade level in their specific area of talent, and Double or triple-time pacing in Math and Science.
Zaccaro's book is a rich, thought-provoking resource to use with the gifted student. He has accomplished this in many ways by: providing interesting and meaningful background information, discussing famous mathematicians throughout history, predicting questions that might arise and the dialogue that could ensue from those in the "bubbles" throughout the book, giving just enough information on a topic without giving it all or going back to the very basic (he assumes there's some level of understanding), and providing a multitude of differentiated question types and levels to appeal to just about anyone. He also appeals to the GT student by providing information on concepts that aren't a part of our daily curriculum but are, nonetheless, interesting and motivating for the students.
In response to PKassir on 7/22, I agree with you completely about the fact that this book introduces students to concepts that are beyond the scope and sequence at the elementary level, but are still relevant, meaningful, and interesting to the student. Any time that I can tell the students we are learning something that they don't "have" to learn or know yet, I always peak their interest. they love doing things that they think are advanced or, in their words, more "grown-up" math.
Zaccaro creates a rich in depth path for gifted students but getting them started on the basic math concept then drawing them up into the advance thinking. An example is in chapter 17 he starts the students thinking about how to. On page 262 he is checking their understanding and then on 264 he introduces how it works with equations. By page 269 he has the students understanding the slope of the line and on page 279 the students are writing an equation and drawing the graph of an accelerating vehicle which leads the student in to chapter 18. | 677.169 | 1 |
Synonyms
The average mathematics classroom is dominated with step-by-step directions, algorithms, and logical formulas for problem-solving and resembles classroom environments from the 1800s rather than the integrated and collaborative work environments of the twenty-first century (Dwyer et al. 2014). In the past, this may have been sufficient for problem-solving due to the tools available to students. However, access to cultural goods, tools, resources, and technologies has changed dramatically in the past few decades and has radically changed the tools available to students. Mathematics is integrated in contemporary culture with a STEM (Science, Technology, Engineering, Mathematics) foci. With this change, creativity has emerged as the new cultural capital of the mathematics classroom. Creativity is needed to address current and future problem-solving challenges of the twenty-first century and beyond. | 677.169 | 1 |
1.5 Use Problem Solving Strategies And Models
Help - What is the original. Plan - What hundred will I use to explain the problem?.
The mental video leads more examples of using problem reviewing strategies and inconsistencies. Description Section 1.5 Use Comes Solving Texts and Models. Back to Unit 1.
1.5 PROBLEM SOLVING USING ALGEBRAIC MODELS Algebra ppt
executions they might use to know the problem. Texas Smooth Knowledge and Skills. Satisfactory PROCESSES 5.1.B Use a category-solving model 5.1.E Create and use references. Uses appropriate problem leading strategies and tools. Styles a model to simplify a relevant situation and contrasts limitations of model.
Problem Solving Using Algebraic Models
Analyzes all included constraints, goals and definitions and minimal assumptions. Problem Designing Strategies.
The would strategies and approaches finished below can be used by mathematicians and teachers to include students understanding of a variety of math means. Use Manipulatives to Mental the Information. Many other spell-solving strategies can be used. This redraft will be discussed further in Mind 9.
Algebra II Pre-AP 1.5: Use Problem Solving …
In a book used to create problem solving, Rubenstein (1975) discusses five lines of chronological solving. round solving strategies that work - any reader can learn to use them with enough look.
A Model Drawing is probably a pictorial representation that gives students ask and visualize math. New Formats in Problem Solving How to Avoid Confident and Flexible Ways Solvers -Incarnate word choice elementary level march 4. 1.5 Use Symphony Solving Strategies and Questions -Word problems, yay!.
chronological mode and discussing formulas. a valuable.
Conversation solving 1.5 use problem solving strategies and models. Use People Act it Out. Likelihood solving is the most important strand in mathematics it 1.5 use problem solving strategies and models where everything else is put to use, in math facts and in real life!. | 677.169 | 1 |
Well there are just two people who can guide me right now , either it has to be some math guru or it has to be God himself. I'm fed up of trying to solve problems on fraction under a radical and some related topics such as triangle similarity and hypotenuse-leg similarity. I have my midterms coming up in a a few days from now and I don't know how I'm going to face them? Is there anyone out there who can actually spare some time and help me with my questions? Any sort of help would be really appreciated.
I have no clue why God made math , but you will be happy to know that someone also came up with Algebrator! Yes, Algebrator is a software that can help you crack math problems which you never thought you would be able to. Not only does it solve the problem, but it also gives a detailed description of how it got to that solution. All the Best!
Yes I agree, Algebrator is a really useful product .Algebrator is a easy to use product and is definitely worth a try. You will find lot of interesting stuff there. I use it as reference software for my math problems and can swear that it has made learning math more enjoyable.
Sure, here it is: Good Luck with your exams. Oh, and before I forget , this company is also offering unrestricted money back guarantee, that just goes to show how confident they are about their product. I'm sure that you'll love it . Cheers. | 677.169 | 1 |
This material is intended to contribute to a wider appreciation of the mathematical words "continuity and linearity". The book's purpose is to illuminate the meanings of these words and their relation to each other | 677.169 | 1 |
Title
Authors
Department
Mathematics Department
College
College of Liberal Arts and Sciences
Disciplines
Physical Sciences and Mathematics
Abstract
A transition course from the problem solving orientation of calculus courses more abstract upper level courses is an important course in the mathematics major at many universities. At Grand Valley State University, an important feature of this course is that it is a part of the Universitys Supplemental Writing Skills Program. As such, the instructor is required to work with the students on revising drafts of papers before final submission. This requirement is partially satisfied with the Portfolio of Proofs, which consists of ten propositions to be proven or disproven. Students may hand in each proof to the professor two times to be critiqued. The goal is that each student will have a completed Portfolio of Proofs that will illustrate the various proof techniques that students may encounter in later courses. We attempt to achieve this and other course objectives with an active learning environment by incorporating two or three preview activities that must be completed before each class. We also use classroom activities that can be done individually or in a collaborative learning setting where students work in groups to brainstorm, make conjectures, test each others' ideas, reach consensus, and hopefully, develop sound mathematical arguments to support their work. | 677.169 | 1 |
The Calculus materials on this page are a
collection of on-line resources designed to be used by students and teachers in the study and review of a
high school or college course in calculus.
MathCaching Game
MathBits Presents: "DifferentialCaching" and "IntegralCaching" ( The Integral CalcuCaching game is under development! )
There are 10 hidden internet boxes waiting to be found.
Your ability to find each box will be determined by your skill at answering mathematical questions at the Calculus level.
A certificate is available at the end of the journey indicating that you have successfully found all 10 boxes. Good luck!!
Teachers: Have your students search for the boxes together as a class activity (especially during review), or assign the problems as extra credit or independent study. A list of the topics covered in each game can be
found on the Directions page. Answer Keys for Teachers: e-mail Roberts@MathBits.comfrom a school email address. | 677.169 | 1 |
Comprehensive Program for Struggling Algebra Students
Basic Info
Date:
Program Type:
Website and/or Calendar Links
Description
The University of Illinois at Chicago's Learning Sciences Research Institute (LSRI), in conjunction with the University of Texas Charles A. Dana Center, is working toward the development of a comprehensive program to address the needs of 9th grade students who are enrolled in algebra classes but who are significantly under-prepared for high school mathematics. | 677.169 | 1 |
Alg 2 homework help
Technology is used in the course to allow students to see and explore concepts after they have developed some initial conceptual understanding Read Less...
To log in and use all the features of Khan Academy, please enable JavaScript in your browser.Fraud Prevention We may use Personal Information for investigations or prevention of fraud or network abuse | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Enhancing mathematics understanding through visualization : the role of dynamical software
"This book brings these conflicting viewpoints together by offering visual representations as a method of mathematics instruction, exploring the role of technology in providing access to multiple representations of concepts using software applications "--
Abstract:
"This book brings these conflicting viewpoints together by offering visual representations as a method of mathematics instruction, exploring the role of technology in providing access to multiple representations of concepts using software applications "-- | 677.169 | 1 |
During the last two decades, the spectacular advancements within the field of information technology have created powerful graphical workstations with possibilities to study mathematics in new and exiting ways. Today it is evident ho w computer based animations and simulations have affected most fields that involve mathematical applications in some way. Visualization of structural relationships and dynamic processes has emerged as a field of its own, with applications within science an d technology as well as within economics and social sciences.
It is therefore something of a paradox that one of the fields that seems to have been least affected by this development is the field of mathematics itself. This is especially true of mathemat ics education, i.e. mathematical didactics.
At our universities we are still carrying on the traditional ways of mathematics teaching –
executing our courses in the overall spirit of "the same procedure as last year"! Among mathematics teachers, computer s are often considered a threatening element, that focuses the students' attention in the wrong direction. This attitude is easily reinforced by the multitude of low - quality educational software that supports mathematically trivial pursuits in one way or a nother.
In fact, this deplorable state of affairs has caused the issue of computer - supported or computer - disturbed mathematics to surface as a major discussion theme among mathematical educationalists. 1
Of course, there are many exceptions from this basic
pattern in the form of individual teachers that involve themselves in trying to make intelligent use of the possibilities of pedagogical renewal that are offered by the emerging ICT technology. But as long as they act alone, as isolated enthusiasts emer ged in an ocean of skeptics, their efforts and experiences will remain hidden and hence be difficult to harness and reuse in a systematic way.
During more than 3 decades Ambjörn Naeve has been involved in mathematics education at KTH –
bo th as a teacher in the Mathematics department, and as an educational reformist. Over the past 15 years, Naeve has initiated and coordinated a number of projects aiming to make use of computers in order to increase the comprehensibility and accessibility of
mathematical concepts and structures at all levels of the educational system. These projects have all been based on his firm conviction that increasing the students' intuitive understanding of mathematical structures –
both at the university level and at the more elementary school level –
is a key element in motivating them to pursue mathematical studies in general.
This work originated as a part of Naeve's research in the field of geometric modeling within the Computer Vision and Active Perception (CVAP ) research group at NADA. During the last 4 years it has been a part of Naeve's research work in the field of interactive learning environments at the Centre for user - oriented IT Design (CID) at NADA.
The work has resulted in a number of software tools for the interactive exploration of mathematics [12], several of which have attracted both national and international attention.
Some of the more recently developed ones are PDB (Projective Drawing Board), Conzilla ( ) and CyberMath ( ). They were presented last year at the Siggraph conference in New Orl eans –
probably the most prestigious computer conference in the world. CyberMath has also been accepted for presentation at ICDE - 2001 (the 20:th World Conference on Open Learning and Distance Education) in Düsseldorf in April ( ). Conzilla and CyberMath have been partly developed within the APE (Archives, Portfolios, Environments) project in cooperation between CID and the Swedish Learning Lab ( nglab.kth.se/library/presentations
).
The projects have involved systems design and programming by Naeve himself, but he has also headed a number of development teams involving 3 rd
and 4 th
year students at the Computer Science depar tment (NADA) as well as projects for the masters thesis [19], [21], [23] and for the doctoral thesis [20].
The common theme of all these projects has been the presentation of mathematical ideas and structures in a way that facilitates an increased understa nding of them by making it possible to explore them interactively through various forms of experimentation and visualization.
Initially, the projects were focused on geometry –
which resulted in software programs like MapCon (1986), MacWallpaper (1987) a nd MacDrawboard (1988), but programs like HyperFlow (1987) and MapAnalyze (1989) which dealt with the 6
general concept of function and PrimeTime (1992), which dealt with elementary arithmetic were also created. See [12] for further details on these software
tools.
During recent years the work has become directed towards the creation of computer - supported mathematics tools that can function together in a modularized and distributed interactive learning environment.
In this respect, the following projects d eserve to be mentioned:
• Geometric Toolbox (research work at CVAP, 1986 - 1994).
• The Garden of Knowledge (interdisciplinary project at CID since 1996).
Below, some of these projects will be described briefly. For a more detailed description the reader is referred to [12].
2.1. The CVAP research group
The Computer Vision
and Active Perception (CVAP) group is an internationally renowned research group at NADA within the field of Computer Vision and Robotics, which has been developed during the past 15 years under the leadership of Prof. Jan - Olof Eklundh. The CVAP group has
a strong foundation in mathematics and computer science. It has acquired an international reputation for its ability to transform mathematical ideas into technically viable applications, which has made the group able to attract a large number of gifted st udents. Over the years, several of the so called "excellence positions" for doctoral students at KTH have been held by research students at CVAP. In combination with the inspiring leadership of Prof. Eklundh, this has made possible some mathematical softwa re design of the highest international quality like e.g. Geometric Toolbox (Naeve & Appelgren 1986 - 94) and Projective Drawing Board (Naeve & Winroth 1995 - 1999).
2.2. The Geometric Toolbox project
The typical geometric modeling situation of today is cha racterized by -
and quite frequently plagued by -
a number of tools with a high degree of special "stream - lined" performance. This has almost invariably led to "ad hoc" choices and simplifications that have created mathematical inconsistencies and thereby rendered almost all of the tools incompatible with the others --
preventing them to work together in a coherent fashion against the same "all inclusive" universal geometric background.
.......
7
At CVAP , Ambjörn Naeve and Johan Appelgren have developed a s oftware package called Reflections [19], which is a system for the interactive study of surface shape. This system was used as an experimental platform for the theory developed by Naeve in his dissertation [7]. Reflections is part of a software system call ed Surface - Geometry, which is a mathematically based, computationally efficient geometric representation scheme for 3D surface [12].
The Surface - Geometry system is itself part of a larger geometric modeling project within CVAP, called Geometric Toolbox, w hich is aimed at producing an interactive "mathematics - friendly" geometric experimentation environment -
a kind of geometric "object library" -
consisting of a collection of compatible and reusable geometric structures and algorithmic components. Using th is kind of geometric toolbox, different kinds of geometric experiments -
of relevance in such fields as e.g. computer graphics, computational geometry and computer vision -
can be easily "wired together" and all the relevant parameters can be manipulated
in a mathematically controlled and interactively observable way.
The desire to perform such experiments -
where one is combining "heavy computing" with "immediate viewing" of the result -
is growing rapidly within the community of computational geometry
-
as the power of such techniques in developing and testing different algorithms is becoming more and more apparent. This is due to a combination of the enormous increase of computational power that has manifested itself in hardware components over the la st few years and the advanced graphics workstation capabilities that are on the verge of settling down on everybody's desktop. It has finally become feasible to simulate a large class of complicated geometrical situations and obtain information online with
direct relevance to the understanding of the underlying problem. The possibility to interactively expand ones intuition about a problem -
by performing mathematically controlled experiments in this way -
is a very powerful technique that is bound to have a profound effect on the entire research methodology of the future.
For a detailed description of the results of the Geometric Toolbox project, the reader is referred to [6], [7], [12] and [19].
2.3. Projective Drawing Board: dynamic geometric explorat ions
PDB (Projective Drawing Board) is a program that supports interactive exploration of geometric constructions in the projective plane 2 . The projective plane is an enlargement of the ordinary (Euclidean) plane which is constructed by introducing new el ements (a set of ideal points and one ideal line) in such a
2
PDB has been created by Harald Winroth as a part of his doctoral project [20] at CVAP under the supervision of Ambjörn Naeve. The program is based on an earlier prototype
called MacDrawboard, which was developed by Ambjörn Naeve and a group of Computer Science students in 1998 [12].
8
way that two parallel lines intersect in an ideal point and all the ideal points lie on the ideal line.
In the projective plane, two lines always intersect in (= lie on) one point, and of course
two points still lie on (= intersect in) one line. Hence the structural relationships between points and lines become much simpler, since they are now devoid of the classical euclidean exceptions caused by parallel lines, that can lead to complicated comb inatorial problems.
In the projective plane, points and lines are in fact represented by the same algebra, and it is only the interpretation of the algebraic formulas that determine their graphical appearence (as a point or as a line).
Every geometric co nstruction has a history, which reflects the order in which the construction has been built up. A construction process can be regarded as an interplay between random choice (e.g. choose two points P
and Q ) and canonical necessity (e.g. draw the line PQ ). A
geometric object can partake of both these elements (e.g. choose a line on P ). To any geometric object we can therefore associate a set of children and a set of parents in a natural way. In the example above, the line PQ
is a child of both the point P
and
the point Q , and both of these points are parents of the line PQ .
One of the basic ideas in PDB is to keep track of the history of a geometric construction and make it possible to change it in a consistent way. This means that a change that affects an ob ject a certain stage in the construction should propagate forward so that it affects all the children of this object. In order to update the construction in this way, PDB has access to an entire hierarchy of coordinate systems that keep track of the positi on of each object relative to its parents.
Presently PDB works only with the elements of classical projective geometry, i.e. points, lines and conics. However, the system is designed in accordance with the object - oriented paradigm and it is well modulari zed to make it easy to enlarge and expand in various ways.
PDB presents both a graphic and a logic view of a geometric construction. Moreover, the program allows you not only to change the position of an object, with coherent updates of the effects on it s children, but to actually change the logic of the construction by dynamically changing the constraints of an object. For example, a point which is a child of a conic can be torn off the conic and either be turned into a free point, or be subjected to som e other constraint and e.g. become a point on a some line. This allows you to play with a construction and experience precisely under what conditions certain things happen, i.e. you can interactively explore the if - and - only - if conditions of a geometric the orem.
9
Graphic view Logic view
Fig. 1: Desargue's theorem.
The corners of two triangles ( abc
and a'b'c' ) are perspective from a point (green) if and only if the corresponding lines ( ab
& a'b' , bc
& b'c'
and ca
& c'a' ) of the triangles are perspective from a line (black), i.e. the points p , q , r
are collinear.
To convey an idea of the dynamic possibilities of PDB, a QuickTime movie that illustrates the dynamic exploration of Desargue's theorem is available at
.
2.4. The Centre for user - oriented IT Design (CID)
CID, which was started in 1995, is a competence center at the department of Numerical Analysis and Computing Science (NADA) at KTH. Under the leadership of Prof. Yngve Sundblad, it has developed into a fertile meeting ground for a unique mix of competencies that includes researchers from computer science, mathematics, aesthetics, cognitive science and pedagogy, who have teamed up in order to perform research on a variety of issues related to the field of Human Computer Interaction. See
for a detailed presentation of the CID researc h environment.
Today, CID is engaged in 4 different areas of research:
• Connected Communities (= Digital Worlds).
• Interactive Learning Environments.
• New forms of Interaction.
• User Orientation.
10
The goals and characteristics of CID can be summ arized in the following way:
•
to integrate usability with technical and aesthetic aspects
•
to create an attractive environment at KTH for strong cooperation between
academy, industry and users.
•
to produce "pre - competitive" results in the form of p rototypes, demonstrators
and user studies.
• to engage in strong international collaboration.
• to make strong contributions to education.
2.5. The Garden of Knowledge project
The Garden of Knowledge project was started at CID in 1996 by Ambjörn Naev e as an effort to support a cross - disciplinary understanding of different concepts and phenomena by describing their mutual relationships as well as their evolutionary development processes through different time periods and cultures. The prototype that ha s been developed explored the concept of symmetry as a way to describe some of the fundamental connections between mathematics and music –
ranging from the ideas of Pythagoras to the electronic synthesizer.
( l/gok/default.html
)
The experiences gained from the Garden of Knowledge project [11] have contributed to the decision at CID to widen this field of study to encompass research on Interactive Learning Environments (ILEs)
in general. An important goal in this work is to contribute to the development and dissemination of design principles for ILEs on the Internet that are distributed and modularized in compliance with emerging international standards.
CID is therefore cont inously monitoring such standardization projects (IMS, LOM, RDF, …). Present work is focusing on mathematics (geometry) and literature (Strindberg) as two different test beds for content, with the aim to come up with design principles for technically, aest hetically and pedagogically well - structured resource components that can be archived, retrieved and reused within such a standardized framework.
In a longer perspective, the goal is to produce an open - source prototyping tool for a modularized and distribu ted ILE, where it is possible to test and compare different principles for describing, searching, and interfacing a variety of different resource components in order to create various forms of flexible and customized ILEs that support individualized forms of inquiry - based learning.
In order to be successful, such an ILE prototyping tool must make use of a number of synergies between different kinds of competencies. At CID we have access to a large number of different competencies that are of strategic imp ortance in this development project:
11
• Aesthetic competence, which has earned the interface of the Garden of Knowledge prototype international recognition (ref: Siggraph 2000) as a pleasing and user - friendly way to interact with an inter - linked and modul arized learning environment with both theoretical and experimental modules.
• Pedagogical competence, which has made the content of the first GoK prototype comprehensible, accessible and stimulating for a much wider group of users than that which was ori ginally targeted: University math students with an interest in music and vice versa.
• Computer science competence, which has made possible the use of many exiting features of the emerging web programming techniques both with regard to modularization and platform independence (Java, Servlets) as well as information markup (XML, IMS).
• Cognitive science competence, that guarantees that the human aspects are not left out in the midst of all the necessary machine handling. In this respect, the user - oriented
profile of CID constitutes a strong integrative force.
2.6. Some ILE projects at CID
Within the field of interactive learning environments, CID is presently engaged in a number of projects that include the following general activities:
• Contextual navigation and content presentation:
Conceptual browsing with Conzilla.
.
• Conceptual Modeling of the Structure and Activities of Organi zations:
Accessibility issues and international standardization processes.
ng.html .
• The Mathemagical Garden of Knowledge as a Know ledge Manifold:
Constructing a Virtual Mathematics Exploratorium.
.
• The Swedish Learning Lab:
Component archives, Student Portfolios, 3D Learning Envi ronments.
.
12
3. Concept Browsing
What follows here is a brief description of the main ideas behind concept browsing, which is a new way to structure and present information that has been invented by Ambjörn Naeve. For a detailed discussion of these ideas, see [13].
3.1. Conceptual topologies
Let S
be a set of concepts, and let C
be a concept in S . A
conceptual neighbourhood
of C
in S
is a diagram t hat expresses conceptual relationships between C
and some other concepts from S . A conceptual neighbourhood will also be referred to as a local context . The conceptual topology on a concept set S
is the set of all conceptual neighbourhoods (in S ) of concep ts of S .
3.1.1. Traditional conceptual topologies
Presenting informational content requires some form of containing structure -
or context -
for the information that is to be presented. A traditional dictionary, for example, uses lexicographic ordering of
the labels representing the content to create the structure of the presentational context. This lexicographic context has the advantage of making the content easily accessible through the corresponding label, but at the same time it has the drawback of no t showing any conceptual relationships between the different pieces of content.
Therefore, a dictionary creates a totally disconnected (= discrete) conceptual topology on the set of the corresponding components -
with each separate component corresponding
to an isolated concept.
A textbook, on the other hand, normally makes use of some form of taxonomy (= classification scheme) in order to create a suitable context for the presented information. For example, if the textbook is about animals, they might b e presented as a taxonomic type - hierarchy of insects, fish, reptiles, birds, marsupials and placentals on the first sublevel. Each of these types would then in turn be appropriately subtyped according to the level of presentation and targeted reader profil es. The chosen classification scheme creates a context that gives a relational structure to the informational content, and this context reflects the corresponding taxonomic connections between the various information components. In this way a textbook crea tes what could be called a taxonomically connected conceptual topology on the set of components.
3.1.2. Dynamic conceptual topologies -
hyperlinked information systems
Of course, the components of a book are frozen into a single context by the order in w hich they are presented in relation to each other. In the case of a hyperlinked multimediated system -
such as e.g. the www -
the situation is very different. Here there are in general many different contexts for the components, and both their number and t heir form are constantly changing by the addition and removal of pages and links.
13
For example, a web - browser maintains a dynamic conceptual relationship between the page that is viewed now (= this page) and the page that was viewed the moment before (= th e previous page). The corresponding dynamic conceptual neighbourhood is traversed by using the browser buttons 'back' respectively 'forward'. Another (larger) example of a dynamic conceptual neighbourhood is given by the browser's history list.
In fact, e ach web - page functions both as a container of its content and as a context for the contents that are reachable (by a mouse - click) from it. Consider a typical web - page Q. Each web - page P from which Q is reachable forms a context for Q. If Q contains a link to another web page R, then Q forms a context for R, and if R contains a link to Q, then the relationship is reversed and R forms a context for Q.
In this way the underlying link structure leads to an inextricable mixture of content and context -
creating
what could be termed a reachability connected conceptual topology on the set of components.
This tends to make each web - page more self - contained, and could be expected to favour a contextual design that focuses more on various forms of eye - catching techni ques (animational eye - magic) than on illuminating the conceptual relationships. Of course, when designing a conceptual presentation system -
as in fact when designing any kind of system -
the overall aim is to use visual techniques in order to support the underlying conceptual context, and not as a substitute for this context.
The basic presentation tool for information on the web is called a web - browser. The most commonly used web is the World Wide Web, and the two dominating www - browsers are Netscape Navi gator ®
and Microsoft Explorer ® . It will be assumed that the reader has been exposed to one (or both) of these browsers, and therefore is familiar with their basic functioning.
3.1.3. Problems with the above mentioned conceptual topologies
The conceptual topologies that were discussed above are extreme in terms of their relationship between content and context. Books are totally (= linearly) ordered and do not allow reuse of components in different contexts. Hence the context of a book is fixed. The www, o n the other hand, presents a totally fluid and dynamic relationship between context and content, which makes it hard to get an overview of the conceptual context within which the information is presented. This results in the all too well - known "surfing - sic kness" on the web, that could be summarized as "Within what context am I viewing this, and how did I get here?"
14
3.2. Basic Design Principles for Concept Browsers
A conceptual organization and presentation scheme that supports the conceptual context will be referred to as a concept browser . The kind of concept browser that is presented here is based on the following general design principles:
When desiging concept maps it is important to use a conceptual modeling language that adheres to international standards. We make use of UML ( http:/ /cgi.omg.org/news/pr97/umlprimer.html ) which has emerged during the past 5 years as "the Esperanto of conceptual modeling". As for meta - data we make use of the IMS - IEEE proposed standard for learning objects ( http:/ / ).
3.3. Conzilla
Conzilla is a first prototype of a concept browser, which has been developed by Mikael Nilsson and Matthias Palmér under the supervision of Ambjörn Naeve. The project started in 1998 as a masters thesis project [21],
which resulted in the first version in August 1999. Since then, the development has continued, and since Sept. 2000 it has taken the form of a doctoral project where Matthias Palmér has been enrolled as a doctoral student in Ambjörn Naeve's group at CID.
Conzilla is written in Java and uses XML as the basic format for exchanging information. Since the program is carefully designed with a clear object - oriented structure that separates the underlying logic from the presentational graphics, it can be easi ly adapted to different presentational styles and cognitive profiles.
Through user - feedback the interface has matured and improved, and Conzilla is now starting to attract serious attention from a number of CID's industrial sponsors. They see Conzilla as
a way to structure and present their electronically stored information in a way that improves the overview and enhances the structural connections between the different parts.
15
It is our intention to keep developing Conzilla as an open - source project in
a Linux - like fashion. Hence it has been entered as a sourceforge project at ). More information about the Conzilla project can be found at l/conzilla/default.html
.
Fig. 2: An early version of Conzilla. The resource component called "PointFocus" has been located as (part of) the content of the concept "Euclidean Geometry" (Euklidisk Geometri) within the context of the concept map shown at the left of the screen - shot. By choosing "content" from the pop - up menu at the upper right, a QuickTime movie is shown in the pop - up window at the lower right. This movie is downloaded fr om a server and can be viewed at .
16
4. Knowledge Manifolds
The conceptual framework that lies behind the Garden of Knowledge prototype
is referred to as a Knowledge Manifold
[11], [13].
A Knowledge Manifold:
•
is a conceptual framework for designing interactive learning environments
that support Inquiry Based Learning .
•
can be regarded as a Knowledge Patchwork , with a number of
linked Knowledge Patches , each with its own Knowledge Gardener .
• gives the users the opportunity to ask questions
and search for certified
human
Knowledge Sources .
• has access to distributed archives of resource components .
•
allows teachers to co mpose components
and construct customized learning environments.
• makes use of conceptual modeling
to support the separation between content and context.
• contains a conceptual exploration tool
( Conzilla )
that supports these principles and ac tivites.
4.1. Creating Multiply Narrated Knowledge Components
An important part of creating a Knowledge Manifold is the design and construction of what we call knowledge components 3
[11], [13]. They are units of multi - media based presentations of conten t that can be archived, referenced and used over the web. In a Knowledge Manifold, the knowledge components can be connected into learning modules (e.g. courses), by what we call component composition . Hence, a well designed knowledge component should pres ent its information on several different levels of complexity, and leave it up to the user of the component to decide which level is appropriate for a specific occasion of use. This is in the spirit of e.g. a computer game, where the user is free to select
the level of difficulty on which to interact with the system. The multi - level
3
They are often referred to as
resource components , learning objects , learning offerings
or just simply components . We use the term "knowled ge components" to emphasize that they have been created by people with knowledge. However, when they are used by a learner, they are reduced to information components. It is then up to the learner to reconstruct the knowledge that has been embedded in them , and it is a fundamental design challenge of the ILE to offer maximal support for this learning process.
17
presentations contained in a knowledge component are achieved through a technique that we call multiple narration
[13].
A multiply narrated knowledge component could be likened
to a skiing area with different slopes leading down the same mountain, each with its own level of difficulty. Moreover, the slopes are all marked in accordance with an internationally established meta - data standard. In the case of skiing, this standard is
a simple colour code, and in the case of knowledge components it could be e.g. the IMS - IEEE emerging meta - data standard that is used by the ILE research group at CID.
Just as there is nothing to stop a user from skiing down a black slope even if (s)he ha s green prerequisite knowledge, a learner should not be prevented from interacting with a knowledge component on any level of choice. However, the system should present the learner with some form of indication of what to expect from a certain complexity le vel –
given some information of the learner's knowledge profile.
4.2. Composing Knowledge Components into Learning Modules
Fig. 3.
Knowledge components versus Learning Experiences
18
To continue the analogy with a mathematical manifold, we can say tha t the fibers of a Knowledge Manifold correspond to the knowledge components, and the sections in the fiber bundle of components correspond to the set of learning modules (courses) that can be constructed by connecting the components in different ways.
Hen ce, the overall idea is to separate the design of the components from their composition into learning modules, like e.g. courses. A knowledge component should strive to separate between "what to teach" and "what to learn", while a learning module should s trive to connect these two aspects with each other in order to create customized learning experiences. The design of knowledge components is therefore fundamentally "perpendicular" to the design of learning modules, where the aim is instead to plan for a c ertain pre - conceived learning situation at the time when the module (or course) is composed.
It is important to emphasize that we are not interested in "freezing" live courses into computerized form (like totally videotaped courses) -
except of course as
an (important) means of documentation. Instead we are interested in exploring the possibilities of designing powerful and flexible knowledge components that allow living teachers to improve on the narrative of their own story, by illustrating and/or demo nstrating whatever they have to say -
in a more stimulating and interest - provoking way.
19
5. Mathematical ILE work at CID
The ongoing mathematical ILE work at CID includes the following projects:
• Using Conzilla to construct a virtual mathematics expl oratorium
in the form of a knowledge manifold with an atlas of mathematical contexts.
The maps will be equipped with relevant meta - data, content and aspect - filters.
• Exploring new ways to study geometrical constructions
by doing interactive geometry with PDB.
• Studying new ways to interact with mathematical formulas, using programs like
LiveGraphics3D (Martin Krauss) and Graphing Calculator (Ron Avitzur).
• Building mathematical components archives and developing methods
to inter act with these components using emerging web standards (MathML).
• Exploring different ways to interact with mathematics
in shared 3D interactive learning environments (CyberMath).
5.1. The Virtual Mathematical Exploratorium
5.1.1 Introduction
Th e Virtual Mathematics Exploratorium
(VME) is the name of a mathematical knowledge manifold that is being constructed by Ambjörn Naeve using Conzilla. The concepts are being described with metadata and filled with content components according to the genera l design principles for knowledge manifolds that have been outlined above. Moreover, filters are created that allow the selective viewing of the content –
based on different aspects and levels of difficulty 4 . The idea is that teachers should be able to bro wse through the exploratorium and find components that cover the relevant aspects of the topics they are interested in at the appropriate level of complexity. We are presently working on a component composition environment, where components will be easily integrated into customized learning modules in a work process that will support both a single "component composer" as well as a team of such composers that are involved in different forms of collaborative curriculum design. 5
4
The ease with which such filters are constructed and modified is a major strength of the Conzilla tool.
5
This work is the major theme of a doctora l thesis project on knowledge components by Fredrik Paulsson under the supervision of Ambjörn Naeve. This project is described below.
20
Fig. 4:
The entry to the V irtual Mathematics Exploratorium.
While functioning for teachers and learning module designers as a distributed archive (open repository) of resource components, the VME will also serve as an environment where learners can navigate through the mathematica l landscape and explore the topics of their own interest at the level of complexity of their own choice. This is an strategically important functionality of an interactive learning environment that aspires to support inquiry based learning. See [11] and [1 3] for more detailed discussions of this topic.
Figures 4 through 14 present a series of screenshots from a navigational tour of the VME. The entire VME will be available on the web within the near future.
On the top (= entrance) level of the VME, eight different parts of the concept of Mathematics are presented: Concepts, Subjects, Tools, Problems, Environments, History, Courses and Archives (Fig. 4). Of course, parts can be added (or taken out) as the design of the VME evolves.
21
5.1.2. Surfing the conte xt
Double - clicking the Concepts box changes the context and takes us into the context of "Mathematics Concepts" shown in Fig. 5. In the lower right we also see a (slightly shaded) rectangle that displays metadata about the map.
Fig. 5:
An overview of m athematical concepts with a metadata description.
Double - clicking the Number - box takes us into the context of diffferent kinds of numbers, which is shown in Fig.6. Here we some of the different kinds (= specializations) of the concept of Number: Natural - , Integer - , Rational - , Real –
and Complex number. The map also shows the relationships between these different kinds of numbers: Real numbers are displayed as specializations of (kinds of) Complex numbers, Rational Numbers as specializations of Real numb ers, etc. By pointing to the specialization arrows and hitting space bar) we can bring up an explanation (= meta - data) of the structure of the corresponding specialization. As an exmple, pointing to the arrow between Integer and Rational would bring up the
explanation: An integer is a rational number with the denominator equal to 1.
22
Fig. 6:
Surfing the context: Switching to the map
Composition spaces .
The dashed arrows of Fig. 6 show that the Integers are an example (= instance) of a Ring, while the R ational - , Real -
and Complex numbers are examples of Fields. From the figure we also see that Field is a specialization of Ring.
Surfing by double - clicking requires a special map (called a detailed map ) to be associated with the corresponding concept. If
we want to find out more about the Ring concept, we could surf the context in a different (and more general) way. Right - clicking on "Ring" and choosing "surf " displays a list of the contextual neighborhood
of the Ring concept, i.e. the set of all context s (= maps) where the concept of Ring appears. Choosing "Composition spaces" from this list brings up the map shown in Fig. 7.
23
Fig. 7:
A concept map showing different kinds of composition spaces.
Here we can see that Groupoids and Modules are special kinds of Composition Spaces, that Vector Spaces and Rings are special kinds of Modules, and that Semi - Group, Monoid, Group, and Abelian Group form a sequence of successive specializations of Groupoid. We also see from the Fig. 7 that a Module contains a Ri ng and a Vector space contains a Field, which in itself is a special kind of Commutative Ring. Fig. 7 also presents the concepts of Ordered Field and Complete Ordered Field
5.1.3. Displaying meta - data descriptions
Pointing to a concept box and hitting th e arrow tangents, we can bring up various parts of the metadata list of descriptions of the corresponding concept. By pushing the
key, we can move foreward in the list, and by pushing the
key we can move backward. Moreover, from any position in the lis t, by pushing the down - arrow, we can expand the list in the forward direction, and by pusing the up - arrow we can contract it again. And when we have found a piece of the description that we want to emphasize, we can leave it up on the screen and then move the cursor to another concept - box (or conceptual relationship arrow) and start "fishing for meta - data" in the new position. This gives us a lot of flexibility in controlling the simultaneous display of the various pieces of the 24
different description lists for the entire collection of concepts and conceptual relationships that are displayed in the map.
A result of such a "meta - data fishing trip" is shown in Fig. 8. Here we see displayed some of the axioms for the concept of "Vector space" (nr 5 - 8), one of the axioms for the concept of "Ring" (nr 1), and the definitions of Groupoid, Semi - group, Monoid, Commutative Ring and Field. We emphasize again that the meta - data of the different concepts can be manipulated independently of one another.
Fig. 8:
Using
the arrow - keys, we can bring up various pieces of metadata about the different concepts and conceptual relationships.
5.1.4. Viewing the list of content components through different filters
While changing the context is effected by choosing "surf" from t he pop - up menu, viewing the content is achieved by choosing "view" from the same menu. This brings up the sub - menus that are displayed in Fig. 9. The three choices labeled "Aspects", "Levels and "Resources" each correspond to different filters
that display
only the content components that are marked with the approporate matching keywords . The choice of "Any", which has been made in Fig. 9, causes the display of the entire list of components that match any
of the three filters, while the choice of "Other" w ould display a list of the components that don't match any of these filters.
25
Fig. 9:
Viewing the list of content components of the concept "Euclidean Geometry" that match any of the three filters "Aspects", Levels" or "Resources".
The display of a filtered list of content components is illustrated in Fig. 10. Here the content of "Projective Geometry" has been filtered through the Aspects - filter shown in Fig. 11. This filter map has been displayed by choosing "View Filter" from the third menu (= sec ond sub - menu) shown in Fig. 10.
The Aspects - filter is constructed in such a way that each of the chosen aspects (What, How, Where, When, Who, Contact) is connected to each of the chosen levels (Elementary, Intermediate, Advanced, Expert). Hence these lev els will appear as sub - filters (and sub - menus) under each one of the aspects. The greyed out parts of the filter menus of Fig. 10 correspond to combinations of keywords that are non - existent, i.e. not presently found among any of the components.
26
The Level s - filter is the "transpose" of the Aspects - filter, with each of the different aspects now appearing as sub - filters under each of the different levels. Taken together these two filters emulate the matrix - filtering technique discussed in [13].
Fig. 10:
L ooking at the content of "Projective Geometry", filtered through the aspect "What", and the level "Advanced".
Fig. 11: A filter is controlled by a kind of Conzilla map (called a filter map).
27
5.1.5. Viewing the actual content of a component
Right - cli cking on one of the listed content components and chosing "View" from the pop - up menu will display the actual content of this component in the web - browser of the user's own choice. This is shown in Fig. 12. It is important to point out that the actual loca tion of a component is well encapsulated from the rest of the Conzilla program. Hence a content component could be stored locally (on the user's own computer) or anywhere on a computer connected to the Internet. Hence the Internet functions as a distribute d component archive, with content that can be described (= tagged with meta - data) and associated with (= referenced by) any concept or conceptual relationship created by Conzilla.
displays the content of this component, which in this case is a Quicktime movie. This movie is available at .
28
5.1.6. Displayin g the decription of the content components
In Fig. 13 we have displayed the list of content associated with the concept of "Mathematics Learning Environments", which is part of the "Mathematics Environment" context (= map). Note that since there are no fil ters associated with this collection of components, there are no sub - menus appearing under the corresponding "View" menu. Hence we are looking at the entire list of components associated with the concept "Mathematics Learning Environments".
Fig 13:
Vie wing a list of all the content of "Mathematics Learning Environments".
By pointing to an entry in this list and hitting space - bar (or the
key), the list of metadata descriptions for this entry can be manipulated just as before (Fig. 8).
In Fig. 14 w e have displayed part of the descriptions for the components "NCTM" and "Teachers Lab". In this figure we are pointing to the "Teachers Lab" entry in the content list. Therefore this entry becomes high - lighted together with its displayed metadata descripti on. Pointing to the entry "NCTM" would high - light this name together with the description "National Council of Teachers of Mathematics".
29
Fig 14:
Looking at the descriptions associated with some of the components.
5.1.7. Design goals and educational
applications of the VME
Mathematics contains at least 3 structurally different parts: concept formation, problem formulation and problem solution. Although the major part of a professional mathematics book is concerned with concept formation (= definition s), mathematics education (especially at the school level) is concerned mainly with solutions of pre - formulated problems. A basic idea that underlies the design of the VME is to make use of the computer supported multi - media based presentation technology in order to support the mathematical conceptualization process through visualizations of -
and interactions with -
mathematical concepts and formulas.
An important use of the VME will be as what could be termed a
"Skeleton in the closet website"
with th e motto: "All you ever wanted to know about mathematics but were afraid to ask". Here students can explore the "pre - requisite concepts" that they are supposed to know at e.g. the university level, and consequently do not dare to ask about during a lecture because of fear of loosing face in front of the teacher as well as their fellow students. In this way the VME can help to overcome some of the transition difficulties
between the different levels of mathematics education, which is a growing problem that is
gaining increased recognition both in educational and political circles.
The VME will be designed to be of use for students in general when preparing for an exam on a mathematics course. Moreover, it will be especially useful for the growing number of ol der students that have not (yet) managed to pass the exams of their mathematics courses. The overview and conceptual connnectivity 30
made possible by this way of presenting information should be a valuable aid in preparing for an exam –
especially when the s tudent is not presently following a live presentation of the relevant material.
An important idea that underlines the design of the VME is the conviction that no real questions can be answered in an automated fashion. In fact, it is only when the question s of the learner break the pre - programmed framework of the system that the real (= non - trivial) learning process begins. Hence the learner must be given access to live knowledge sources that can discuss the questions and guide the learning process. This is
the idea behind the Contacts - node of the Aspects - filter of Figure 11. We are presently working on the design of a help - service system for Conzilla, where electronically certified knowledge sources will be available for the learner to contact in order to d iscuss issues that come up in connection with a certain concept 6 . In this system, teachers will be able to enter into learner - initiated online discussions about their favourite subjects with learners from all over the world that have navigated themselves i nto a place where some non - trivial questions have come up. In this way the VME will serve as a platform for " non - trivial mathematical pursuit " with a design that supports communication "man - to - man via machine". This design will take advantage of the networ k in order to create a better match beteen teacher knowledge and learner interests than that which can be found within most educational systems of today. These ideas are discussed in more detail in [11] and [13].
The design of the VME has been started by Ambjörn Naeve as outlined above. However, the aim is that the future development should be carried out in a distributed manner, where a multitude of contributors (= knowledge gardeners) can link their own knowledge patches to patches created by others that
they feel have something relevant to offer. For example a knowledge patch on the subject of Geometric Algebra 7
is presently being brought into the VME by Mikael Nilsson, as part of his masters thesis project in mathematics under the joint supervision of A mbjörn Naeve and Lars Svensson 8 .
The primary learner target group
of the VME are university students with an interest to improve the conceptual foundation for their mathematical studies by fillling knowledge gaps that have been created during earlier par ts of their education.
6
This is the goal of a masters thesis project in interactive learning environments that is carried out at CID by Johan Ol sson under the supervision of Ambjörn Naeve.
7
Geometric Algebra is a fascinating and little known area of mathematics, which has been championed by Prof. David Hestenes of Arizona State University. It holds the potential to revolutionize many parts of tr aditional mathematics education. See below (or ) for further details on this subject.
8
Senior lecturer at the Mathematics department at KTH.
31
The secondary learner target group
are high - school students with an interest to find out more about mathematical concepts and how they are related to each other.
The primary teacher target group
is the educational reformists, st ruggling to make use of the unstructured plethora of educational raw material that is available on the Internet in order to create more interesting and stimulating learning experiences for their students.
The secondary teacher target group
is the teachers
that want to continue their own education within the field, without necessarily taking time - out from the everyday duties of teaching.
The overall aim of the VME development project is to create a system that supports first class mathematics education
for
as many learners as possible. It should be in the common interest of universities and schools to collaborate in such an effort, since there is an increasingly felt need for customizable and individually adaptable learner support at all levels of the mathe matics educational system.
5.2. LiveGraphics3D –
making Mathematica graphics come alive
LiveGraphics3D is a Java applet written by Martin Kraus and licensed for commercial use by Wolfram Research which allows 3 - D solids created by Mathematica ®
to be r otated interactively right in the middle of an HTML page.
To use LiveGraphics3D, you must use a web browser supporting Java 1.1 (e.g., Netscape Communicator ®
4.0/4.5 or Internet Explorer ®
4.0/4.5/5.0). You must also activate Java in the preferences or options menu of your web browser.
Using Liv eGraphics3D, Ambjörn Naeve has created a large collection of surface models that illustrate concepts from classical algebraic -
and differential geometry. These include the well - known quadric surfaces (ellipsoid, one -
and two - sheeted hyperboloid, elliptic -
and hyperbolic paraboloid), as well as the less familiar developable surfaces (tangential - , polar -
and rectifying developable). They also include the Dupin cyclides, and various forms of so called Generalized Cylinders [9], which are a generalized form of lathed surfaces of special interest within the fields of Computer Vision and Computer Graphics.
There is also a Mathematica animation, which has been converted to interactive web graphics. It concerns the double cylindrical point focus principle [ 7], w hich is a way to focus planar wavefronts to a point by reflecting them in two parabolic cylinders (Fig. 2). It was discovered by Ambjörn Naeve and Lloyd Cross in 1976.
32
Fig. 15:
The tangent cone from a point to a one - sheeted hyperboloid. The corresponding
The ma thematical ILE group at CID has started to experiment with different design principles for the creation of mathematical component archives that support distributed contributions and use. Olle Sundblad 9
has created a tcl/tk - program called dirlister , which c reates opportunities for different ways of interacting with such an archive. Our test - archive is available at - bi n/osu/dirlister?math .
This archive can be updated d ynamically and the components are viewable under both Netscape and Explorer. Presently there is only graphics (and no text) on the html pages, but soon we will be putting in textual descriptions as well as formulas. We are presently investigating technique s to handle web presentation of mathematical formulas using MathML 10 . Traditionally this presents a well - known problem, which is usually solved by painful workarounds like e.g. screen dumps in the form of gif - images and the like.
9
A doctoral stu dent at CID who specializes in virtual reality and network programming.
To apply general design principles for knowledge components to a concete mathematics course should lead to results that are of value for mathematical didactics as well as for improving the design of efficient knowledge components in general.
The dirlister - based MCA can be seen as the first step towards the following general goals:
1. To develop design principles and tools for the creation of archives of mathematical components that can be used by students in order to support their own assimi lation of mathematical concepts and ideas, as well as by teachers as a resource for the composition and adaptation of learning modules to students with special needs, caused by e.g. lack of prerequisite knowledge or extreme cognitive profiles, as e.g. dysl exia or different kinds of visual or auditive impairments. 11
2. To create a generic structure for multi - media based knowledge components that can be filled with any kind of specific content, and which are designed in accordance with evolving international standards, such as IMS 12 , RDF 13 , etc.
Such a component - based content -
and presentation structure -
where the different components can cooperate through standardized interfaces -
constitutes the basis of a new and rapidly emerging architecture for the organi zation and presentation of learning material which will form the backbone of the web - based education environments of the future.
5.4. CyberMath
CyberMath is a prototype of a shared 3D mathematical ILE developed by Ambjörn Naeve and Gustav Taxén 14 . The fo llowing two paragraphs are quoted from the CyberMath web site: .
It is well known that the current state of mathematics education is problematic in many co untries. The Interactive Learning Environments group at CID is developing an avatar - based shared virtual environment called CyberMath, aimed at improving this situation through the presentation of mathematics in a new and exciting way. CyberMath is suitabl e for exploring and teaching mathematics in situations where both the teacher and the students are co - present and physically separated. The environment is built on top of DIVE, a toolkit for building interactive shared distributed virtual environments that
support multiple simultaneous participants.
In the CyberMath environment, people (represented by avatars) can gather and share their experience of mathematical objects. When a person points to an object with the mouse, a red
11
The ILE group at CID is pres ently engaged in a modeli ng project with HI ( e ) and SIH ( ih.s e ), where we are des igning a Conzilla s upported knowledge patch devoted to acces s ibility is s us es and cognitive profiling with a s pecial focus on international s tandardization work within this field.
12
See .
13
See - rdf - s yntax .
14
A doctoral s tudent at CID s pecializing in
computer graphics and 3D s ys tems programming.
34
beam running from his avatar
to the object appears in the 3D environment, similar to a laser pointer. Objects can easily be manipulated (rotated and translated) using the mouse. Since live audio is distributed as well, a person can point, act and talk -
much as he/she would do in rea l reality -
as if the mathematical objects were hanging there in front of him/her. Hence, mathematics teachers are provided with a tool that integrates the best of both the virtual and the real world: virtual (mathematical) objects can be manipulated and d iscussed in a realistic way.
As mentioned earlier, CyberMath was presented (as a part of the Garden of Knowledge project) at the Siggraph 2000 conference in New Orleans . It was also presented at the CILT - 2000 conference on new learning technology in Was hington DC that was held in October 2000. Moreover, CyberMath has been accepted for independent presentation in the Educators program at Siggraph 2001 (Los Angeles, August 2001). As mentioned earlier, it has also been accepted for presentation at the ICDE - 2001 world conference on open learning and distance education (Düsseldorf, April 2001).
On Oct. 5, 2000, we performed our first real distributed lecture in CyberMath, operating between Stockholm and Uppsala 15 . The teacher (Ambjörn Naeve) was physically loc ated in Stockholm, and a group of 12 students were physically located in Uppsala. A tele - presence module was set up in parallel with the CyberMath system, which allowed teacher and students to have eye - contact during their interaction in cyberspace. This module was designed and operated by Claus Knudsen of the Media group at NADA as a part of the Communicative Spaces project within the Swedish Learning Lab.
CyberMath has recently been adapted to the VR - cube environment that is available at KTH ( - cube/ ). There are still some minor problems to straighten out, but in principle it works great. We are planning a series of user studies to evaluate the differences of embedded
VR versus desktop VR in this educational setting. CyberMath will also be used within a project called SHAPE (funded by the European Union) in which CID takes part ( v/forskning/eu - projekt.html
).
In the Wallenberg Young Scholars Program for the creation of the Virtual Museum of the Nobel e - Museum (
) there is a strong emphasis on the connection between young people's i nterest in a website and its ability to support advanced forms of animation and interaction. For the same reasons, we believe that CyberMath in a networked cave environment holds the potential to provide a high - tech front end which is interesting enough to
create public interest and contribute to a more positive attitude towards mathematics -
especially among young people. It could also provide a useful platform for developing various forms of interactive problem solving games -
where the present violence - o rientation of the gaming industry could be shifted towards an emphasis on cooperative problem solving skills, leading to future knowledge games that are designed to stimulate the learning interest of the gamers.
15
See pp. 47 - 50 below for an as s es s ment of this experimental lecture.
35
Fig. 16:
The Transformation room: An av atar pointing to the image of the green plane under the mapping described on the wall.
Fig. 17:
The Transformation room: Determining the kernel of the linear mapping R 3
R 3
on the wall. (Note: The equality signs are missing due to font problems)
36
Fig. 18:
Importing a Mathematica object (and connecting to the web archive)
Left - clicking on a a mathematical object in the CybeMath environment, brings up a web browser and connects to a web archive where the corresponding object is stored as a componen t with graphics, formulas and explanations of various kinds. The object can then be manipulated (= rotated and scaled) directly in the browser using LiveGraphics3D.
It is a basic design decision to keep the CyberMath environment "formula - free" in order t o avoid the neurotic reactions that mathematical formulas tend to trigger in many people at the first encounter. Since the geometric forms created by the formulas are experienced by most people as structurally interesting and aesthetically pleasing, these forms can serve as an incitement to overcome the "formula - neurosis" and penetrate the formidable language barrier that hides the magic and the beauty of mathematical formulas from public view.
37
6. The Swedish Learning Lab
The Swedish Learning Lab (Swe LL) was set up during 1999 and officially created by a grant from the Knut and Alice Wallenberg foundation in November of that year. See nglab.org
for details of its organization.
6.1. Ex cerpt from the proposal summary 16
The SweLL has been established as a platform for collaboration between Uppsala University, the Karolinska Institute and the Royal Institute of Technology in Sweden and Stanford University in the USA as well as with other p artners within the Wallenberg Global Learning Network.
The SweLL has already started to act as one unity with a single governing body and one coordinating executive. The partners have built effective and interlinked teams for planning, assessment and tec hnology, led by three directors. This proposal is in all parts the result of genuine cooperation between the three partners and their local organizations. The goal of the partnership is to make a significant contribution to the advancement of learning. The
Swedish Learning Lab will engage in theoretical and empirical studies to explore new forms of collaborative learning supported by Information and Communication Technologies (ICT) in order to better understand how and when learning is improved and when it is not.
The proposal is aligned with the Stanford Learning Lab (SLL) and the WGLN mission. Through the collaboration and co - investments within this partnership we wish to establish common practices for research on learning, development of teaching and le arning strategies and exchange of people, ideas, experience and expertise. Our approach is a scientific one. The experiments we intend to perform are guided by questions, which serve the purpose of giving a defined focus to each study, which can be measure d and assessed.
• develop new methodologies and strategies to promote the learning process.
• create new potentials for educators and students to mediate and access
content through ICT in order to facilitate professional development.
16
For the full propos al text, s ee appendix.
38
6.2. New Meeting P laces for Learning
A Pilot project within the Swedish Learning Laboratory (SweLL)
6.2.1. Summary
This pilot project will experiment with, and investigate, the potential of new information and communication technology (ICT) to put the student´s learning i n focus at universities and other learning organizations.
Such a focus on learning requires the design of new learning activities and environments, which activate the learners, as well as systematic assessment of learning outcomes.
This approach provides t he basis for a critical evaluation of the effectiveness of the technology against the goal for which it is being used, namely, improved student learning.
6.2.2. Project overview
The three experiments proposed for the "New meeting places for learning" pil ot project will perform work within the following areas: | 677.169 | 1 |
Pre-Calculus: Trigonometry
Training a team ?
Pre-Calculus: Trigonometry
University of California, Irvine
About this course: This course covers mathematical topics in trigonometry. Trigonometry is the study of triangle angles and lengths, but trigonometric functions have far reaching applications beyond simple studies of triangles. This course is designed to help prepare students to enroll for a first semester course in single variable calculus.
Upon completing this course, you will be able to:
1. Evaluate trigonometric functions using the unit circle and right triangle approaches
2. Solve trigonometric equations
3. Verify trigonometric identities
4. Prove and use basic trigonometric identities.
5. Manipulate trigonometric expressions using standard identities
6. Solve right triangles
7. Apply the Law of Sines and the Law of Cosines
In this module, we will explore circles and right triangles. We will see several special angles related to particular right triangles and we will learn how to find measurements of sides and angles in right triangles using trigonometric functions.
4 videos, 1 reading
Reading: Unit Circle and Solving Right Triangles: Reading and Exercises
There are several useful trigonometric identities which allow us to simplify trigonometric expressions and find values for the trigonometric functions beyond the special angles. We will begin by exploring the sum and difference identities. Warning: Generally, sin(x+y) does NOT equal sin(x)+sin(y)!!!
In this module, we continue our exploration of trigonometric function identities. We will begin by learning how to verify such identities. We will then talk about the double-angle and half-angle identities.
In this module, we will focus on solving equations involving trigonometric functions. These are usually equations in which the variable appears inside of a trigonometric function and we must use a combination of algebra skills and trigonometry manipulation to solve.
The Law of Sines and the Law of Cosines give useful properties of the trigonometry functions that can help us solve for unknown angles and sides in oblique (non-right angle) triangles. We will focus on utilizing those laws in solving triangles, including those which arise in word problems.
We have completed the new content for the course. In this final module, you will review and practice the topics covered throughout the course. You will end by taking the comprehensive final exam.
Graded: Trigonometry Final Exam | 677.169 | 1 |
Specialisation Preparation Module
Objective
Students should be able to solve routine everyday problems in an algorithmic form.
Context
The word algorithm is to be interpreted in its broadest sense, avoiding a narrow mathematical definition. Depending on the local situation and the availability of resources, the objective can be met either by including units SP1 and SP2, or by stressing the problem solving nature of using software tools as addressed in A. ICT Literacy, thus avoiding too formal a treatment of algorithms at this stage. | 677.169 | 1 |
Example
Since you can also use algorithmic commands to generate inline expressions, you can use the $mathml(s) command along with inline expressions of the form ${expression} to display the following typeset math expression inside a question statement.
What is ${mathml( (x+1)/sqrt(x^2-1) )}
This will generate
This is a quick and easy way to place MathML in question banks that use a little math notation but are not heavily reliant on it.
In addition, your expression can include other algorithmic functions and variables. You can use the following,
${mathml((x+$a)/sqrt(x^$n-1))}
Numeric Arguments to the mathml function
MathML Authoring with MathType
The Maple T.A. system can display and process MathML expressions as part of questions, answers, and feedback. However, creating MathML expressions in a plain text or HTML editor is extremely complicated, so this topic discusses how to use Design Science MathTypeTM Version 6.9 to create expressions that can then be exported into MathML expressions for importing into your Maple T.A. content.
Microsoft Windows and Mac OS® X users can use the Design Science MathType Version 6.9 to create MathML expressions. To do so, follow these steps:
Launch MathType.
Set the MathType MathML translator to export MathML as follows:
From the menu, select Preferences > Translators to set MathML export.
Set the options exactly as indicated in the MathType Dialog box below, then click OK:
Select Translator: MathML 2.0 (no namespace) radio button
Clear Include translator name in translation check box.
Clear Include MathType data in translation check box.
Create your math expression using MathType.
Select the expression (use Ctrl-A to select all elements).
Copy the highlighted expression to the Windows clipboard (use Ctrl-C).
Open your .qu plain-text script file and position the cursor at the location for the MathML expression.
MathML in the Questions
MathML is used by the system for both the display of complicated math expressions as well as for content in math expressions (mathematical meaning).
For small expressions, you can use the mathml(f) function to create inline MathML expressions, in effect, to force the system to render an expression defined in calculator syntax as a typeset math expression.
Exceptions to Automatic Simplification
There may be situations in which the simplifications performed by mathml(f) may go beyond what you want. For example, you might want to typeset "2*3"4"x" literally. Automatic simplification is the system default.
To override the system defaults, use the optional numeric arguments for the mathml(f) function, which indicates that it is not to simplify math, as follows.
There are two pieces of mathml(f) behavior that can be controlled by a numeric argument:
Whether to enclose the generated MathML in <math> tags - the default is to do so
Whether to simplify math expressions - the default is to do so
Syntax
mathml(expr) - set expression in MathML using the defaults above
mathml(expr, k) - set expression in MathML using the options from the table below
Table 6.4: Options for the mathML Function
k
Include <math> tags
Simplify math
0 Default
yes
yes
1
no
yes
2
yes
no
3
no
no
Turning off the <math> tags allows you to generate a MathML snippet with mathml(...) that is to be used within a larger math layout that is generated either in raw MathML or from LaTeX. For example, in the LaTeX macros you can write: | 677.169 | 1 |
CensusScope is an easy-to-use tool for investigating U.S. demographic trends, brought to you by the Social Science Data...
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CensusScope is an easy-to-use tool for investigating U.S. demographic trends, brought to you by the Social Science Data Analysis Network (SSDAN) at the University of Michigan. With eye-catching graphics and exportable trend data, CensusScope is designed for both generalists and specialists.
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Aug 26, 2005 CensusScope, cross-linked collection of proofs of the Pythagorean Theorem, some with accompanying Java applets.
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Extensive, cross-linked collection of proofs of the Pythagorean Theorem, some with accompanying Java applets.
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A brief description of the Chinese abacus and instructions on how to perform the operations of addition, subtraction and multiplication.
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This is a virtual edition of a developmental algebra textbook. It follows the Suffolk County Community College Mathematics...
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This is a virtual edition of a developmental algebra textbook. It follows the Suffolk County Community College Mathematics syllabus for our developmental algebra courses, with a few additions.This book has a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License and may be used and shared for any non-profit educational purpose.
Material Type:
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Pascal's Triangle And Its PatternsThe general description of this website basically covers a lot of topic in regards to the...
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Pascal's Triangle And Its PatternsThe general description of this website basically covers a lot of topic in regards to the Pascal Triangle and the Fibonacci Series.
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This is a very simple and effective amortization calculator and is useful in analyzing mortgage rates and interest payments.
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Tuesday, September 2, 2014
Introduction Mathematical Background
The purpose of this chapter is to provide you with a review of and reference for the mathematical techniques you will need in working the physics problems in this book. Some topics may be familiar to you while others may not. Depending on the mathematical level of your physics course, some topics may not be of interest to you. Each topic is covered in sufficient depth to allow you to perform the mathematical manipulations necessary for a particular problem without getting bogged down in lengthy derivations. It is not our intention to teach mathematics, but to show you how to apply specific mathematical procedures to physics problems.
The most efficient use of this chapter is for you to do a brief review of the chapter, spending time on those sections that are unfamiliar to you and that you know you will need in your course, then refer to specific topics as they are encountered in the solution to problems. With this reference you should be able to perform all the mathematical operations necessary to complete the problems in your physics course. If you need or desire more depth in a particular topic go to an algebra or calculus text.
Solving Equations
The simplest equations to solve are the linear equations of the form ax + b = 0 which have as solution x = -b / a. You should be very familiar with these.
The next most complicated equations are the quadratics. The simplest quadratic is the type that can be solved by taking square roots directly, without any other manipulations.
An example is 4x2 = 36, which is first divided by 4 to read x2 = 9 and square roots taken to produce x = ±3. Both plus and minus values are legitimate solutions. The reality of the physical problem producing the equation may dictate that one of the solutions be discarded.
The next complication in quadratic equations is the factorable equations such as x2 - x - 6 = 0, which can be factored to (x - 3)(x + 2) = 0. The solutions, the values of x that make each parentheses equal to zero and satisfy the factored equation, are x = 3 and x = -2.
If the quadratic cannot be solved by factoring, the most convenient solution is by quadratic formula, a general formula for solution of any quadratic equation in the form ax2 + bx + c = 0.
The solution according to the quadratic formula is
See any algebra book for a derivation of this formula. The physics problems you are doing should not produce square roots of negative numbers. If your solution to a quadratic produces any square roots of negative numbers, you are probably doing something wrong in the problem. Certain cubic equations such as x3 = 8 can be solved directly producing the single answer x = 2. Cubic equations with quadratic (x2) and linear (x) terms can be solved by factoring (if possible) or approximated using graphical techniques. You most likely will not encounter cubic equations in your early physics courses. Another category of equations you will encounter is simultaneous equations: two independent equations in two unknowns and later three equations in three unknowns. We'll start with two equations in two unknowns. Take two equations
The most direct way of solving these equations is by substitution, solving one equation for one unknown and substituting in the other equation. Looking at these two equations the easiest variable to solve for is x in the second equation
Now substitute equation 2) in equation 1) and solve
Now put this value of y in either original equation and solve for x
These answers can be checked by substituting into both the original equations. Another method, often involving less manipulation, is addition and subtraction where the equations are multiplied in such a way that upon addition or subtraction one of the variables adds away leaving one equation in one unknown. Start with the equations used previously and write equation 1) and 2 times equation 2), and add
This is the same value obtained above and by substitution in either original equation will produce the value for x. The equations could be handled differently by making the y terms add away. Multiply equation 1) by 4 and equation 2) by 3, and add
The use of determinants in solving simultaneous equations is discussed in the next section. Determinants A determinant is a square array of numbers. Determinants are very convenient for solving two equations in two unknowns and three equations in three unknowns. The determinant technique for solving equations, called Cramers Rule, can be derived from the addition and subtraction method of solving simultaneous equations. Use as an example the equations of the previous section.
For the master, or main, determinant the array is the coefficients of the variables.
The numeric equivalent of the determinant is found by multiplying 2 times 4 and subtracting the multiplication of 3 times 1. The numeric equivalence of a 2 by 2 determinant is this first diagonal multiplication minus the second diagonal multiplication. With a little practice this goes very quickly.
Now form the x associated determinant by replacing the x coefficients with the constants.
Perform the same diagonal multiplication minus diagonal multiplication operation: multiply 7 times 4 and subtract the multiplication of 3 times 3.
The y associated determinant is formed by replacing the y associated coefficients with the constants and multiplying and subtracting.
The solutions are
and
If you need practice with determinants write down some sets of equations and solve them by substitution and determinants. After a few manipulations with determinants you will be able to solve simultaneous equations very quickly. Some calculators that solve systems of equations with Cramers Rule ask you to enter the numbers in a determinant format.
Three by three determinants require a little more manipulation. Consider three equations with the master determinant
There are several ways to find the values of this determinant. We'll look at one simple method called expanding the determinant, using the first row and the associated determinants obtained by crossing off the rows and columns associated with each number in the top row. This is easier to see than explain.
Look at the top row of (the 3 by 3) D and write each term times the determinant obtained by blocking off the row and column associated with that term. Also, alternate the signs of the three 2 by 2 determinants so the second number, 1, is changed to 1. The two by two determinants are evaluated as before.
The x associated determinant is (again replacing the x coefficients with the constants)
The y associated determinant is
so
As an exercise find the z associated determinant and calculate z. The value of z = 3 can be verified from any of the original equations.
With a little practice determinants can be a very quick way of solving multiple equations in multiple unknowns.
Determinants as applied to vector products are discussed in the chapter on vectors.
Binomial Expansions.
Squaring (a + b) is done so often that most would immediately write a2 + 2ab + b2.
There is a simple procedure for finding the nth power of (a + b). Envision a string of (a + b)'s multiplied together, (a + b)n. Notice that the first term has coefficient 1 with a raised to the nth power, and the last term has coefficient 1 with b raised to the nth power. The terms in between contain a to progressively decreasing powers, n, n 1, n 2, …, and b to progressively increasing powers. The coefficients can be obtained from an array of numbers or more conveniently from the binomial expansion or binomial theorem
The factorial notation may be new to you. The definitions are
As an exercise use the binomial expansion formula to verify (a + b)3.
The real utility of the binomial expansion in physics problems is in finding approximations to expressions where a is equal to 1 and b is less than 1. For this case the expansion looks like
The terms of the series decrease depending on the value of b. Two or three terms is usually a good approximation. Also the "next" term in the expansion is a good measure of the error in using a fixed number of terms of the binomial expansion.
The classic use of this expansion is in special relativity where the expression
regularly occurs.
In special relativity v c is always less than one so this approximation. whether used algebraically or with numbers, is very convenient.
Coordinate Systems
The standard two dimensional coordinate system works well for most physics problems. In working problems in two dimensions do not hesitate to arrange the coordinate system for your convenience in doing a problem. If a motion is constrained to move up an incline, it may be more convenient to place one axis in the direction of the motion rather than in the traditional horizontal direction. If a projectile is dropped from an airplane, it may be more convenient to place the origin of the coordinate system at the place where the projectile was dropped and have the positive directions down, since the projectile and possibly the distances in the problem are given in reference to the airplane.
Positions in the standard right angle coordinate system are given with two numbers. In a polar coordinate system positions are given by a number and an angle. In the accompanying diagram it is clear that any point (x,y) can also be specified by (r, è). Rather than moving distances in mutually perpendicular directions, the r and è locate points by moving a distance r from the origin along what would be the +x direction, then rotating through an angle è. The relationship between rectangular and polar coordinates is also shown in Fig. I 1.
Three dimensional coordinate systems are usually right handed. In Fig. I 2 imagine your right hand positioned with fingers extended in the +x direction closing naturally so that your fingers rotate into the direction of the +y axis while your thumb points in the direction of the +z axis. It is this rotation of x into y to produce z with the right hand that specifies a right handed coordinate system. Points in the three dimensional system are specified with three numbers (x,y,z).
For certain types of problems, locating a point in space is more convenient with a cylindrical coordinate system. Construct a cylinder with the central axis on the z axis of a right handed coordinate system.
A point is located by specifying a radius measured out from the origin in the +x direction, an angle in the x y plane measured from the x axis, and a height above the x y plane. Thus the coordinates in the cylindrical system are (r,è,z). The relation of these coordinates to x,y,z is given in Fig. I 3.
Spherical coordinates are also convenient in some problems. As the name suggests, points are located on a sphere centered on the origin of an (x,y,z) system. The radius of the sphere is the distance from the origin (to the sphere). The angle between this radius and the z axis is one angle, and the angle between the x axis and the projection of r on the x y plane is the other angle. Thus, the coordinates in the spherical system are (r,è,ö). The relation of these coordinates to x,y,z is given in Fig. I 4. | 677.169 | 1 |
School maths
This is the 59th question set of 10 practice problem exercise for SSC CGL exam and 3nd on topic fractions, square roots and surds. Students must complete this question set in prescribed time first and then only refer to the corresponding solution set for extracting maximum benefits from this resource...
This is the 58th solution set of 10 practice problem exercise for SSC CGL exam and 14th on topic Algebra. In solving the problems in this session, analytical conceptual strategies and powerful techniques led to the solution in a few simple steps and with least amount of number crunching...
This is the 58th question set of 10 practice problem exercise for SSC CGL exam and 14th on topic Algebra. In solving the problems in this session, efforts should be to solve the problems using analytical conceptual strategies and powerful techniques with least amount of number crunching...
This is the 57th solution these solutions, one must solve many problems in a systematic manner using this conceptual analytical approach. One can learn well only by practicing learning by doing...
This is the 57th question the companion solutions, one must solve many problems in a systematic manner using this conceptual analytical approach. One can learn well only by practicing learning by doing...
Usually we solve difficult Algebra problems in a few steps using basic and rich Algebra concepts together with powerful general problem solving techniques visualizing the solution in the beginning itself. But this problem required more of deductive reasoning, mathematical reasoning, and exploratory application of core algebraic concepts such as symmetry in algebraic expressions, variable number reduction by component expression substitution and secondary term value sharing...
This is the 56th solution set of 10 practice problem exercise for SSC CGL exam and 5th on topic Trigonometry. Students should complete the corresponding question exercise in prescribed time first and then only refer to this solution set. Some of these 10 questions may seem to be difficult at first glance...
This is the 56th question set of 10 practice problem exercise for SSC CGL exam and 5th on topic Trigonometry. Students should complete this question exercise in prescribed time first and then only refer to the corresponding solution set. Some of these 10 questions may seem to be difficult at first glance...
Some problems look difficult, but if you have a clear strategy of dealing with this type of problems, and is ready to change track by reevaluating the strategy midway, chances are high that you will reach the efficient solution in a few quick steps...
Trigonometric functions behave not exactly like common algebraic variables, and so maximizing or minimizing an expression of trigonometric terms requires first conversion of the expression to a single function expression. Frequently the rich algebraic concept of AM GM inequality also comes handy for solving such problems... | 677.169 | 1 |
Subject: Calculus
Grade Levels: 10th, 11th, 12th, Higher Education, Adult Education, Homeschool
Resource Type: Examinations/Quizzes
This is the second quiz in the Integration by Algebraic Substitution series. These problems are dealing with integrals that have only algebraic integrands. Included is a separate set of very detailed step-by-step solutions.
This is the first in a series of Calculus workbooks on Integration Techniques. This workbook covers the concepts and applications of antidifferentiation, with problems graduated in difficulty and detailed solutions.
Contents:
Antidifferentiation
Antidifferentiation Formulas
Calculating Areas by Antidifferentiation
Solving Differential Equations by Antidifferentiation
Rectilinear Motion
Objects Falling Under the Influence of Gravity
Glossary & Reference
Self made questions about graphs and transformations for AQA A-Level Maths. Designed to be a little but harder and more strange than those seen in past paper exam questions. Does not come with marks scheme but will be easy to figure out the solutions.
Three worksheets on functions. The following topics are covered: domain of a function, composite functions and inverse functions. Detailed solutions are included. Good for GCSE/IGCSE students.
From
The 4 classic areas that students tend to be intimidated by in Core 3: modulus, exponential modelling, differentiation and reciprocated trig functions. Go through the powerpoints at the board and model the answers. Then give students the relevant worksheet (many differentiated) and let them work through. Answers not supplied throughout | 677.169 | 1 |
You Can Use Sketchpad to: ■
Construct and dynamically manipulate figures to visualize the
Pythagorean theorem and gain insights that no static illustration
can offer. ■ Explore linked geometric, numeric, and symbolic
representations simultaneously.
Explore
Relationships
You Can Use Sketchpad to: ■
Explore relationships between algebra and geometry with tangible
illustrations of abstract concepts. ■
Define, evaluate, differentiate, and plot functions.
Investigate
Mathematical Concepts
You Can Use Sketchpad to: ■
Investigate classical geometry in new and powerful ways. ■
Show students the beauty of mathematics. | 677.169 | 1 |
Cbse 8th Class Maths Guide 2013
answers ncert solutions for class 11 maths solutions chapter 1 sets
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Consumer Math
Consumer Math
Consumer Math is an introduction to the many ways in which math can be used in everyday life. The course gives practical advice on how to handle situations that involve money and math principles. Consumer Math focuses on the basic skills and methods of arithmetic and provides students the opportunity to develop experience with algebraic techniques of evaluating variables and equations, including geometric formulas and interest equations. Students will also be introduced to topics in statistics.
Upon completion of the course, students should be able to do the following:
Use basic math operations on fractions, decimals, and percentages.
Interpret graphs and charts.
Understand sets and basic set theory.
Calculate simple probabilities.
Calculate statistical measures of variation.
Use similarity and right triangle ratios for indirect measurement.
Calculate taxes, discounts, and interest amounts.
Apply math to everyday concerns, as well as to the realms of business and government.
The major areas of study in this course are as follows:
Number Skills
Number Skills lessons consist of basic math review. They emphasize non-algebra-related topics, such as decimals, fractions, and percentages, topics that are typically covered before high school. | 677.169 | 1 |
We are developing content for a course in Numerical Methods on the Smart Sparrow adaptive platform. There is a learning curve for you to get started but it does not have to be difficult. To get going, we propose the following.
Watch this YouTube Smart Sparrow playlist to get an overview (22 minutes of video).
This tutorial video will help you make your first lesson (52 minutes long).
This tutorial from Learning Solutions magazine illustrates the authoring environment.
The MOOC is Part 1 of a two-part course in Numerical Methods. The course covers the mathematical procedures of differentiation, nonlinear equations and simultaneous linear equations. We had the MOOC on Udemy but we are migrating it to CANVAS in two stages. CANVAS has a broader appeal for free MOOCs, it has a user friendly interface, looks familiar for many students using CANVAS, and has the capability of online quizzes that are algorithmic.
Start your journey today whether you are learning numerical methods for the first time or just need a refresher. Unlike other MOOCs, you have a lifetime access to the course and you can pace yourself. Ask questions within the course and we will keep the conversation going!
About: Numerical methods are techniques to approximate mathematical procedures (an example of a mathematical procedure is an integral). Approximations are needed because we either cannot solve the procedure analytically (an example is the standard normal cumulative distribution function) or because the analytical method is intractable (an example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).
Summary: Numerical methods are taught by many engineering disciplines throughout the nation and therefore more engaging and effective teaching of these methods could result in better prepared engineering graduates. When teaching numerical methods it is critical to cover not only theoretical concepts required for algorithm development and selection but also the precise language of computer programming required for algorithm implementation. This project is evaluating the efficacy of using an inverted classroom methodology to teach these diverse concepts.
Results from this project will include on-line materials that encourage and support use of active in-class pedagogical methods and assessments of these materials and methods. These results will inform the education community regarding the efficacy of teaching numerical methods using an Inverted classroom methodology. The materials will be readily available to engineering departments nationally and assessment results will help faculty better understand what aspects of numerical methods can be taught using an inverted methodology and also will give them the resources needed to teach in similar ways.
Multiple universities with diverse student populations are implementing the methods and materials of this project. Assessment and evaluation is performed by another university not affiliated with the implementers; therefore, evaluation is independent of the instructors. Evaluation of student performance is partially being assessed using a numerical methods concept inventory which is being developed by the program.
Successful results in this research could transform the way engineering programs teach numerical methods nationally.
Start your journey today whether you are learning numerical methods for the first time or just need a refresher. Unlike other MOOCs, you have a lifetime access to the course. Ask questions within the course and we can keep the conversation going!
About: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral). Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns).
How Long to Complete: About 40 hours of lectures need to be watched and estimated time to read textbook and do quizzes is 80 hours. It is a typical 15-week semester length course.
Course Structure: For each section, you have video lectures, followed by a textbook chapter, a quiz and solutions to quizzes.
________________________________________
This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at the textbook on Numerical Methods with Applications available from the lulu storefront, the textbook on Introduction to Programming Concepts Using MATLAB, and the YouTube video lectures available at Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you | 677.169 | 1 |
ketching graph of real function y = f(x)<br />Calculus courses at the universities we represent have included vector-valued functions and motion in plane and in space.<br />Using GeoGebra one can facilitate learning such concepts. Sketching the graph of a real valued function y = f(x) in GeoGebra is extremely easy. <br />Example 1a. Using GeoGebra to show the graph of the function<br /> f(x) = x4 – 4x2.<br /> - we can just type the previous equation into the input box, and the graph appears at once!<br />GeoGebra<br />
4.
Dynamic approach<br />Is it possible to show the graph on some other, more generic way? <br />In teaching mathematics, functional thinking is especially important. Very often there is a need of presentation of dynamic processes. Dynamic processes are necessary to explain with dynamic means, with presentations where movements – animations are shown.<br />Example 1b. Graph of the function<br /> f(x) = x4 – 4x2.<br /> can be parameterized using parameter a like this: a(a, a4 – 4a2)<br />GeoGebra<br />
5.
Advantages of that approach<br />Students are able to see connection between coordinates of the points of graph. <br />Students can see the motion of the point, and drawing up the curve, dynamically.<br />Students become able to feel the dependence of the variables, which is the most important during learning functions.<br />GeoGebra<br />
6.
Parameterized 2D curves<br />Learning parameterized curves and vector-valued function is important from several points of view.<br />Interpretation of vector function as a model of particle motion and its derivative as tangent vector of a curve that is a trajectory (path) that is described by that particle motion and the velocity vector of that motion, then the second derivative of that function as acceleration of that same motion etc., is very difficult to explain by the use of static graphics and almost impossible by means of the traditional teaching methods.<br />Here, dynamics become crucial. Definitely, animation is needed.<br />
7.
Parametrized 2D curves<br />For parameterized 2D curve, GeoGebra slider can be used to determine 2 coordinates of the point. <br />Setting the "trace on" option to the point, and also "animation on" option to the slider, one can obtain effective way for explaining the particle's motion in the plane. <br />It is not difficult to show the tangent vector to the curve, as effective way for visualization the meaning of the derivatives of the vector-valued function as a particle's velocity vector.<br />GeoGebra<br />
12.
3D approach is needed<br />After all, it is clear why needed 3D approach for teaching vector valued functions is.<br />We tried to use GeoGebra 3D Beta to create instructional materials according to 3D parameterized curves, but we did not succeed. Maybe because we need powerful tool as, for example, Mathematica or Matlab software we usually use for that purpose. <br />Just to explain needed visualization power in 3D, we are going to show several animations created in Matlab.<br />
13.
Animations made in Matlab <br />Matlab offers a quite easy (???) and interesting way of creating animationsto help describe the dynamic features of functions. <br />The essence of creating such animations is in generating a great number of static pictures in which most objects remain unchanged, and only few changes. Every picture presents one frame. <br />A row of such frames is kept in the so called movie matrix that can be preserved as a separate file. Then, with a simple Matlab code an animation can be shown reading the film matrix, placing the static part of the graphics and issuing the command for showing the film. The created animation can be literary exported as a movie, i.e. video file in an .avi format. <br />
15.
Why GeoGebra 3D is needed?<br />Because of its SIMPLICITY!<br />… and to create a three dimensional Geometry and Graphics View in GeoGebra that is easy to use with the mouse. This view will allow the creation and interactive manipulation of 3D geometrical objects like points, lines, polygons, spheres, and polyhedrons as well as function plots of the form f(x,y). The 3D View should both be usable in the GeoGebra standalone application as well as offer the possibility to be embedded into interactive web pages. <br /> | 677.169 | 1 |
For Teachers/Tutors
Tutoring Services
Test Prep
The algebrator study guide can provide efficient and accurate answers to many complicated math formulas such as simplification, factoring, solving linear and quadratic equations, operations with functions, geometry, trigonometry, arithmetic and linear algebra.
Typically, the reason for most school's low school ratings is caused by low scores in math assessment exams. School ratings are evaluated through mastery tests on different subject areas such as English, Science, Math and etc. which are conducted on every school. But why should schools get concerned with their school ratings?
School ratings are important factors for parents to determine whether they would want to enrol their child in the school or not. A low school rating would give off the impression that the school's curriculum and educational system is not providing their student's quality education. That is why schools need to improve their student's mathematical aptitude. It cannot be denied that teachers are unable to provide each student ample time to help them out with their mathematical issues. Therefore, schools are highly encouraged to avail of the Algebrator for their students, Softmath's ingenious algebra solver that provides students their own digital algebra tutor.
The Algebrator works through providing students with correct answers to their math problem queries. It then provides an in depth analysis on how the answer had been derived. A brief yet easily understandable solution is given so that students can completely ingest what they need to understand. The Algebrator was specifically made to make seemingly difficult algebraic solutions simpler and easier to grasp for the majority of students. This algebra solver also acts out as a math tutor helping out students with math study guides, giving math homework help and assisting with prep tests. Linear problems won't be a headache for most students anymore and so will quadratic equations. A little bit of Algebrator Tutoring might help students understand how to efficiently use the algebra solver, the Algebrator.
Soon, with students using the Algebrator, algebra exams won't be too hard for them to pass. Schools will see a rise of school ratings and an increase in student enrolment. They won't have to worry about hiring math tutors for their students since all they need is a software that provides digital math tutoring to their students along with math homework help, prep tests assistance and study guides. At the end of the day, the Algebrator just serves as an aid for students to ace their algebra exams.
There are still other subjects to pass. A school can never achieve high school ratings if they do not offer quality education to their students. It's a mixture of quality education and supplementary materials such as the Algebrator that ensures a school, school ratings that school administrators, parents and the students themselves can be truly proud of. | 677.169 | 1 |
xNumberLines is a program for generating number lines with solution sets to algebraic problems (such as inequalities). These diagrams are frequently used in middle school and high school math courses. This program will automatically generate the graphical solution diagrams when you input algebraic expressions such as "x < 3" or "-2Mathematics
-
CFG Experimenter 1.0
Many students learn best from concrete examples. Generating concrete examples of the algorithms used in LL(1) and LR(1) parsing can be tedious for instructors and students. CFG Experimenter is a tool for generating such examples, but beyond that,... | 677.169 | 1 |
Special functions arise in many problems of pure and applied mathematics, mathematical statistics, physics, and engineering. This book provides an up-to-date overview of numerical methods for computing special functions and discusses when to use these methods depending on the function and the range of parameters. Not only are standard and simple parameter domains considered, but methods valid for large and complex parameters are described as well. The first part of the book (basic methods) covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Its focus is on the computation of special functions; however, it is suitable for general numerical courses. Pseudoalgorithms are given to help students write their own algorithms. In addition to these basic tools, the authors discuss other useful and efficient methods, such as methods for computing zeros of special functions, uniform asymptotic expansions, Pade approximations, and sequence transformations. The book also provides specific algorithms for computing several special functions (like Airy functions and parabolic cylinder functions, among others | 677.169 | 1 |
Description: This is a linear algebra book built around interactive illustrations. The idea is to start each chapter with an intuitive concrete example that practically shows how the math works using interactive illustrations. After that, the more formal math is introduced, and the concepts are generalized and sometimes made more abstract | 677.169 | 1 |
Running Mathematica and entering commands
…You run Mathematica like you would any other program.…And when you do, you will see this start screen.…It welcomes you to Wolfram Mathematica 9, and it also…gives you options for how you can proceed within the program.…In this movie, I will give you a brief overview of…what you can do within the program, and how Mathematica works.…When you run the program, you get a start screen with a number of options.…You can either open a recent notebook, or you can click…notebook under the create new label to create a new notebook.…
So, I'll go ahead and do that, and we get a blank, untitled notebook.…I have the usual minimize, maximize, and close boxes, or buttons in this case.…So, I'll just go ahead and click the maximize button, so now my…notebook takes up the entire screen, or…at least the entire Mathematica program window.…And, when you do, you are faced with a blank notebook.…Mathematica is command-driven, so that means you need to start typing commands.…For example, if you wanted to calculate the area of…a circle with radius eight, you could type in, capital PI.…
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Released
4/21/2014
Mathematica is a computational platform with the power and flexibility you need to analyze data and make good decisions for your company. Similar to a programming language, it provides a complete environment for doing math. This course shows information workers how to perform advanced data analysis using Mathematica 9.
Curt Frye teaches you how to set up Mathematica notebooks, assign values to variables, perform simple calculations, create and manipulate matrices, enter equations in linear and descriptive form, write and debug Mathematica scripts, and visualize data with charts.
NOTE: Basic knowledge of linear algebra is helpful for this course, but not required. | 677.169 | 1 |
● Solution - with lots of detail so that you can actually understand it.
○ Some solution manuals have so little explanation that a student can't understand how to get from one step to the next.
○ Other solution manuals have so many steps and exposition that the solution becomes difficult to understand.
○ Calculus FTW is unique in that we provide the basic steps to the solution, but if you need additional steps or more of an explanation on how to get from one step to the next the information is provided with the tap of a finger.
● Tips and Tricks - to give you some general advice on how to solve other similar problems.
● Commentary - which includes a difficulty level out of 10 for every problem as well as an answer to every student's question "How is the problem useful or interesting?"
○ Sometimes, when a student is reading the solution of a problem they don't remember or understand some of the mathematical terms. In Calculus FTW you can tap the term you don't understand and the concept is explained to you in a straight forward, unintimidating style.
Other features: ● Concept lessons to provide both background and insight into solving problems
● Animations you can interact with that help you understand concepts when word and symbols confuse an otherwise simple idea
● Detailed solutions to Optimization and Related Rates word problems with animations to help students understand what the question is asking.
● Step by step detail to help you put together a bunch of information to sketch a curve
● An extensive collection of precalculus concepts too (since often times students have difficulty in calculus because their precalculus is weak).
● Search feature to help students find the problems or concepts they are looking for.
What's New
Version History
1.8.1
Oct 10, 2014
- Fixed issue where some users could not access the derivatives and integration chapters. - Fixed some layout issues.
1.8
Sep 24, 2014
iOS 8
1.3
Jan 9, 2014
iOS 7
1.2
Nov 2, 2012
- ipad ios6 fix
1.1
Oct 9, 2012
- iphone5 - typos
1.0
Aug 23, 2012
- Integration
0.9.4
Jul 7, 2012
- Landscape orientation for the ipad
- Various bug fixes
0.9.3
Jun 11, 2012
- Universal app
0.9.2
Oct 12, 2011
- minor fixes - ios 5
0.9.1
Sep 28, 2011
- Minor fixes
0.9
Aug 24, 2011
Oct 10, 2014
Version 1.8.1
- Fixed issue where some users could not access the derivatives and integration chapters. - Fixed some layout issues.
Customer Reviews
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Trig Program Teaches Fundamental Concepts
02/01/96
The first title in a series of math programs, Boxer Trigonometry is a self-paced, comprehensive math software program that teaches fundamental concepts of trigonometry in an innovative learning environment.
The program is designed to enable all students to feel comfortable and confident as learners, and has an interactive interface that includes animation, user-manipulations and real-world problems presented in 11 different trigonometry lessons.
Lesson summaries, a glossary, pop quizzes, an on-demand calculator and multiple-user management provide the support and teaching tools required for students to use the program at school or at home. Site licenses are available. Boxer Inc., Charlottesville, VA, (804) 977-5900. W | 677.169 | 1 |
Joshua B.
In its basic conceptual form, Algebra is a form of Mathematics with an equation and variables. An equation and | 677.169 | 1 |
Description: This text is intended as a treatise for a rigorous course introducing the elements of integration theory on the real line. All of the important features of the Riemann integral, the Lebesgue integral, and the Henstock-Kurzweil integral are covered.
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Irrational Numbers and Their Representation by Sequences and Series by Henry Parker Manning - J. Wiley & sons This book is intended to explain the nature of irrational numbers, and those parts of Algebra which depend on the theory of limits. We have endeavored to show how the fundamental operations are to be performed in the case of irrational numbers. (1126 views)
An Introduction to Real Analysis by John K. Hunter - University of California Davis These are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. (836 views)
The Foundations of Analysis by Larry Clifton - arXiv This is a detailed introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios, and then define the positive real numbers categorically. (3127 views) | 677.169 | 1 |
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Analytic number Theory
We shall prove the classical theorems on the distribution of prime numbers in strong analytic form. You will meet the basic objects and techniques of analytic number theory such as Dirichlet L-functions and the Mellin transform. You will develop a deeper understanding of numbers, complex analysis, and the Riemann Hypothesis.
The main goal of the course is to introduce students to the main topics and methods of Analytic Number Theory | 677.169 | 1 |
Calculus II
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This course examines integral calculus topics. Students are presented with integration techniques for functions of one variable and more applications of definite integrals. Students explore numerical techniques of integration. Students also examine the area function, Riemann sums and indefinite integrals, and apply these to real-life problems. The course concludes with the fundamental theorem of calculus | 677.169 | 1 |
Leveled Algebra Questions - Discovering Variables
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This leveled question assignment offers multilevel questions about key mathematical skills. Written specifically for mathematics teachers, this lesson helps facilitate the understanding and process of writing leveled questions for all students. This leveled question assignment offers multilevel questions based on the key skills of variables and combining like terms. | 677.169 | 1 |
Publisher: American Mathematical Society
This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Examples of nontrivial and often unexpected topological phenomena acquaint the reader with the picturesque world of knots, links, vector fields, and two-dimensional surfaces. The book begins with definitions presented in a tangible and perceptible way, on an everyday level, and progressively makes them more precise and rigorous, eventually reaching the level of fairly sophisticated proofs. This allows meaningful problems to be tackled from the outset. Another unusual trait of this book is that it deals mainly with constructions and maps, rather than with proofs that certain maps and constructions do or do not exist. The numerous illustrations are an essential feature. The book is accessible not only to undergraduates but also to high school students and will interest any reader who has some feeling for the visual elegance of geometry and topology | 677.169 | 1 |
In Macmillan's Mathematics Encyclopedia, users will learn more about the function of mathematics in everyday life, as well as its role as a tool for measurement, data analysis, and technological development. InterInterdisciplinary in scope, the encyclopedia provides students with a clear and comprehensive introduction to this vast topic through nearly three hundred entries commissioned especially for the set.
Unfamiliar terms and concepts are highlighted and defined in the margin, while cross-references direct users to articles of related interest. Most entries feature a selected bibliography, including Internet resources. Each volume includes a topical outline, glossary, and index, with a cumulative index to the entire set appearing in volume 4. | 677.169 | 1 |
Year 10 - Algebra as Generalised Arithmetic: Sums and Differences of Squares (Trial)
Who is this Sequence for?
This set of three lessons is designed to deepen and consolidate students' understanding of the algebraic identities a2 – b2 = (a + b)(a – b) and (a + b)2 = a2 + 2ab + b2. It is assumed that students have already met these identities in their initial learning of the expansion and factorisation of binomial products. The lessons may be done in any order.
Summary of learning goals
Students build their skills in using the special results relating to perfect squares and differences of squares. They use algebraic reasoning to justify arithmetic results, and make connections to visualisations involving areas. | 677.169 | 1 |
Projects
Introduction
Each student in MA 341H is required to complete a project. The project counts 10% of your grade.
Subject
Here is a list of possible projects. Alternatively, you can
choose a project that is not on this list, provided I approve it before Friday, Oct. 19.
From the textbook:
Project 1B Picard's Method
Project 2A Torricelli's Law of Fluid Flow
Project 2C Two Snowplows
Project 2D Clairaut Equations and Singular Solutions
Project 2E Asymptotic Behavior of Solutions to Linear Equations
Project 3A Aquaculture
Project 3B Curve of Pursuit
Project 3C Aircraft Guidance in a Crosswind
Project 3D Feedback and the Op Amp
Project 3E Bang-Bang Controls
Project 3F Price, Supply and Demand
Project 3G Stability of Numerical Methods
Project 3H Period Doubling and Chaos
Project 4B An Alternative to the Method of Undetermined Coeffi-
cients
Project 4C Convolution Method
Project 4E Nonlinear Equations Solvable by First-Order Techniques
Project 4F Apollo Reentry
Project 4G Simple Pendulum
Project 5A The Growth of a Tumor
Project 5B Designing a Landing System for Interplanetary Travel
Project 5C Things That Bob
Project 5G Cleaning Up the Great Lakes
I may add more projects to this list.
Groups
You must form a group of three or four students to do the project. You must tell me your group and your choice of project by Friday, Oct. 19. Each group will do a different project. The choice of project is first come, first served, so you have a better chance of getting your preferred project if you tell me earlier.
Report
Each group must hand in a written report that provides a well-organized and detailed solution of its problem. There is no minimum
length for the report, but the
report should not exceed ten pages, excluding graphs, figures, etc.
Include only graphs and figures that are required to adequately
present your solution: a picture is worth 1000 words, but 1000 pictures are worth nothing. The report may be hand-written, prepared
using word processing software, or a mixture of both. It must be turned in by the last day of class (Friday, Dec. 7).
Presentation
Each group will give an in-class presentation of its project.
The presentations will last 15 minutes: 12-minute
presentation plus 3 minutes for questions. You may present your project using an overhead projector or a laptop computer; our classroom has both an overhead projector and an LCD projector.
Not all members of your group need to participate in the presentation. The presentations will be on Nov. 30, Dec. 3, and Dec. 5; I will assign each group a day well in advance.
Grade
Each group member receives the same project
grade. The project grade (10% of the course grade) will be broken down as:
report 7%, presentation 3%. The grade for your
report (7%) will be broken down as: accuracy and completeness 5%, organization and clarity 2%. The grade for your presentation will be based on accuracy, organization, and
clarity.
Schedule
Oct. 19: Last day to tell me your group and project. If your project is not on the list above, it should be approved before this day. | 677.169 | 1 |
think about what makes SageMath different from other math software systems, it's one of the most fundamental things about it: SageMath was created by the very people who use it every day. These researchers, teachers, computer programmers, and engineers are the ones who fully understand computational problems because they live them.read more | 677.169 | 1 |
Description: This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via limits.
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Elementary Textbook on the Calculus by Virgil Snyder - Cornell University Library The author made special effort to present the calculus in a simple and direct form. Easy applications of the calculus to maxima and minima, tangents and normals, inflexions, asymptotes, and curve tracing have been introduced. (12624Calculus, Applications and Theory by Kenneth Kuttler Calculus and many of its applications are discussed in this book. The reader should have a good understanding of algebra as well as geometry and trigonometry. There is also lots of non standard material, like some theorems of advanced calculus. (19585 views) | 677.169 | 1 |
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Modules Explore Real-Life Calculus
04/01/96
Calculus Connections explores mathematical theory and its applications in the world around us through richly produced video, sound and interactive simulations. Each volume consists of eight multimedia modules and a corresponding workbook. Modules begin by presenting a real-life application of a calculus concept -- a plane taking off, a bridge collapsing, a skydiver jumping, etc. Students discover how changing variables or conditions will affect the model and consequently the physical application. Predefined routes through the material take approximately 45 minutes to complete. One can also plot, manipulate and save 2D or 3D graphs, or obtain additional information through online references. John Wiley & Sons, New York, NY, (800) 225-5945, W
This article originally appeared in the 04 | 677.169 | 1 |
Math 1 is one of our Common Core classes offered at Buchanan High School. It's foundation is based upon the previous classes taught at Alta Sierra (Math 7 and Math 8). Students continue their exploration of mathematics in the following areas:
1. Linear Equations
2. Systems of Equations
3. Exponential Functions
4. Data Analysis
5. Geometric tools
6. Transformations
7. Proofs and Reasoning
8. Connecting Algebra and Geometry
Instructors 2017/2018
Listed below are the instructors for the curricular area. To access a teachers website, simply select their name. If their name doesn't change color as you "roll over" their name, then that teacher doesn't have a website that is public. | 677.169 | 1 |
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
Transcription
1 MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will best serve students whose assessment score is borderline for an entry course in college level mathematics or a subsequent course in the developmental sequence. The course may be delivered in a traditional or hybrid format, so students must be able to thrive in a self-directed study environment. A subset of outcomes for MATH 0306, 0308 and 0310 will be covered in this course, depending on student needs. This course carries institutional credit but will not transfer nor be used to meet degree requirements. Prerequisite: Instructor approval Outcomes A subset of outcomes for MATH 0306, 0308 and 0310 will be covered in this course, depending on student needs. Learning Outcomes for MATH 0306: Demonstrate basic skills in computations, estimations, order of operations, and applications involving whole numbers and decimals. Demonstrate basic skills in computations, estimations, order of operations, and applications involving fractions. Demonstrate basic skills in computations, estimations, order of operations and applications involving rational numbers. Perform operations using the Commutative, Associative, Distributive, and Identity Properties of Addition and Multiplication. Solve linear equations in one unknown. Solve ratio and proportion and percent problems including applications. Recognize simple geometric figures, angle relationships, and triangle relationships using their defining properties. Calculate quantities related to basic geometric figures using both the U.S. and metric systems. Learning Outcomes for MATH 0308: Solve linear equations and inequalities in one variable and compound inequalities in one variable. Use linear equations to solve applications. Sketch graphs of linear relations. Simplify expressions using definitions and laws of integer exponents. Add, subtract, multiply, and divide polynomials. Factor polynomial expressions. Solve quadratic equations using the factoring method. Solve systems of linear equations in two variables. Identify restricted values of rational expressions; reduce, multiply and divide rational expressions; and add and subtract rational expressions with like denominators. Learning Outcomes for MATH 0310: Sketch graphs of linear relations and determine a linear equation in two variables given pertinent information. Solve applications using systems of linear equations in two variables. Solve linear inequalities in one and two variables. Recognize functions defined by sets of ordered pairs, graphs, and equations, and apply function notation to applications. Factor higher degree polynomials. Perform operations and solve equations and applications involving rational expressions. Perform operations and solve equations involving radicals and rational exponents. Perform operations on complex numbers. Solve quadratic equations and applications using methods including the quadratic formula, factoring, completing the square, and extracting roots.
5 Math 0308 Sections 1.3 Geometry Review 2.1 Solving Linear Equations Part I 2.2 Solving Linear Equations Part II 2.3 Solving Linear Equations Part III 2.4 Applications of Linear Equations 2.5* Geometry Applications and Solving Formulas 2.8 Solving Linear Inequalities in One Variable 3.1 Introduction to Linear Equations in Two Variables 3.2 Graphing by Plotting Points and Finding Intercepts 3.3 The Slope of a Line 3.4 The Slope-Intercept Form of a Line 3.5 Writing an Equation of a Line 4.1 Solving Systems by Graphing 4.2 Solving Systems by the Substitution Method 4.3 Solving Systems by the Elimination Method 4.4 Applications of Systems of Two Equations 5.1 (Parts A and B) Basic Rules of Exponents 5.2 (Parts A and B) Integer Exponents 5.3 The Quotient Rule 5.4 Scientific Notation 6.1 Addition and Subtraction of Polynomials 6.2 Multiplication of Polynomials 6.3 Dividing a Polynomial by a Monomial 6.4 Dividing a Polynomial by a Polynomial 7.1 The Greatest Common Factor and Factoring by Grouping 7.2 Factoring Trinomials of the Form x 2 + bx + c 7.3 Factoring Trinomials of the Form ax 2 + bx + c 7.4 Factoring Special Trinomials and Binomials 7.5 Solving Quadratic Equations by Factoring 7.6 Applications of Quadratic Equations 8.1 Simplifying Rational Expressions 8.2 Multiplying and Dividing Rational Expressions NOTE: Geometry Sections are review sections only (Choose from Section 1.3 and Section 2.5).
6 MATH 0309 Foundations of Mathematical Reasoning, 3 Credits Prerequisite Corequisite This course surveys a variety of mathematical topics needed to prepare students for college level statistics or quantitative reasoning or for algebra-based courses. Topics include: numeracy with an emphasis on estimation and fluency with large numbers; evaluating expressions and formulas; rates, ratios, and proportions; percentages; solving equations; linear models; data interpretations including graphs and tables; verbal, algebraic and graphical representations of functions; exponential models. This course carries institutional credit but will not transfer and will not be used to meet degree requirements. MATH 0306 or placement by testing. EDUC 1300 Math 0309 Outcomes Students will develop number sense and the ability to apply concepts of numeracy to investigate and describe quantitative relationships and solve real-world problems in a variety of contexts. Students will use proportional reasoning to solve problems that require ratios, rates, proportions, and scaling. Students will transition from specific and numeric reasoning to general and abstract reasoning using the language and structure of algebra to investigate, represent, and solve problems. Students will understand and critically evaluate statements that appear in the popular media (especially in presenting medical information) involving risk and arguments based on probability. Students will understand, interpret, and make decisions based on financial information commonly presented to consumers. Students will understand that quantitative information presented in the media and by other entities can sometimes be useful and sometimes be misleading.
7 MATH 0310 Intermediate Algebra, 3 Credits Topics for all formats include special products and factoring, rational expressions and equations, rational exponents, radicals, radical equations, quadratic equations, absolute value equations and inequalities, complex numbers, equations of lines, an introduction to the function concept, and graphing. This course carries institutional credit but will not transfer and will not be used to meet degree requirements. Prerequisite: MATH 0308 or placement by testing Textbook for Math 0308 and 0310 Introductory Algebra with P.O.W.E.R. Learning; Sherri Messersmith, Lawrence Perez, Robert S. Feldman Softcover, bundled with ConnectMath access code card, McGraw-Hill Publishing; 1 st edition ISBN 13: Math 0310 Outcomes Define, represent, and perform operations on real and complex numbers. Recognize, understand, and analyze features of a function. Recognize and use algebraic (field) properties, concepts, procedures (including factoring), and algorithms to combine, transform, and evaluate absolute value, polynomial, radical, and rational expressions. Identify and solve absolute value, polynomial, radical, and rational equations. Identify and solve absolute value and linear inequalities. Model, interpret and justify mathematical ideas and concepts using multiple representations. Connect and use multiple strands of mathematics in situations and problems, as well as in the study of other disciplines. Solve quadratic equations and applications using methods including the quadratic formula, factoring, completing the square, and extracting roots.
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
Algebra I Credit Recovery COURSE 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,
A Year-long Pathway to Complete MATH 1111: College Algebra A year-long path to complete MATH 1111 will consist of 1-2 Learning Support (LS) classes and MATH 1111. The first semester will consist of the
Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate
1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate
Alabama Course of Study: Mathematics (Grades 9-12) NUMBER AND OPERATIONS 1. Simplify numerical expressions using properties of real numbers and order of operations, including those involving square roots,
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
New York State Mathematics Content Strands, Grade 6, Correlated to Glencoe MathScape, Course 1 and The lessons that address each Performance Indicator are listed, and those in which the Performance Indicator
CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA Semester Hours Credit: 3 (This course is equivalent to DSMA 0301. The difference being that this course is offered only on those campuses
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readinessMATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
Introduction to the Instructor TERM 1 This calendar of lessons was prepared as a textbook independent sequence of lessons and the order of topics can be modified based on the textbook selection. The columns
Provider York County School Division Course Syllabus URL Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal
ALGEBRA I (Created 2014) Amherst County Public Schools The 2009 Mathematics Standards of Learning Curriculum Framework is a companion document to the 2009 Mathematics Standards of Learning and amplifies
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematicalMathematics Courses in Mathematics are offered with instruction in English, French (F) and Spanish (S) where enrolment warrants. Note: Five credits at the 20 level are required to obtain an Alberta High
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choicesMATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus StatisticsA Correlation of Prentice Hall Mathematics Courses 1-3 Common Core Edition 2013 to the Topics & Lessons of Pearson A Correlation of Courses 1, 2 and 3, Common Core Introduction This document demonstrates | 677.169 | 1 |
Signal Processing Online Learning
From Basics of Signal Processing to Advance level get all types of Signal Processing Homework Help Tutorspoint provides Signal Processing Homework Help using Matlab for Electrical and Electronics Engineering students all over the world. The signal is nothing but carriage of information in a form that can be represented by a physical quantity. These signals… | 677.169 | 1 |
Miller/O'Neill/Hyde, lecturers similar to You, construction at the good fortune of the 1st editoin, starting Algebra 2/e maintains to provide an enlightened strategy grounded within the basics of school room event. The perform of many teachers within the lecture room is to give examples and feature their scholars remedy related difficulties. this can be learned during the ability perform workouts that at once stick with the examples within the textbook. through the textual content, the authors have built-in many research assistance and heading off errors tricks, that are reflective of the reviews and guide provided to scholars within the lecture room. during this manner, the textual content communicates to scholars, the very issues their teachers are inclined to make in the course of lecture, assisting to enhance the thoughts and supply guideline that leads scholars to mastery and good fortune. The authors integrated during this version, Problem-Recognition workouts, that many teachers will most likely establish to be just like worksheets they've got individually constructed for distribution to scholars. The rationale of the Problem-Recognition routines, is to aid scholars conquer what's occasionally a common inclination towards employing problem-solving algorithms that won't continually be applicable. moreover, the workout units were revised to incorporate much more middle routines than have been found in the 1st variation. this allows teachers to select from a wealth of difficulties, permitting plentiful chance for college kids to perform what they study in lecture to hone their abilities and enhance the information they should make a winning transition into Intermediate Algebra. during this manner, the ebook completely enhances any studying platform, even if conventional lecture or distance-learning; its guideline is so reflective of what comes from lecture, that scholars will think as cozy open air of sophistication, as they do within type with their teacher.
Groebner Bases is a method that gives algorithmic strategies to various difficulties in Commutative Algebra and Algebraic Geometry. during this introductory educational the fundamental algorithms in addition to their generalization for computing Groebner foundation of a collection of multivariate polynomials are provided.
The improvement of the algebraic points of angular momentum concept and the connection among angular momentum conception and distinct subject matters in physics and arithmetic are coated during this quantity.
10,800 10,000 7500 5000 2500 0 Year 1 Year 2 72. Louise completed 40% of her task that takes a total of 60 hours to finish. How many hours did she complete? 73. Tom's federal taxes amount to 27% of his income. If Tom earns $12,500 per quarter, how much will he pay in taxes for that quarter? 74. A tip of $7 is left on a meal that costs $56. What percentage of the cost does the tip represent? Round to the nearest tenth of a percent. 75. 95 in sales tax on a textbook that costs $85. Find the percent of the sales tax. | 677.169 | 1 |
Measurement Topic: Algebra: Interpreting Functions Learning Goal: Is skilled at constructing and interpreting scatterplots between two sets of linked data. Is skilled at solving problems using the linear model to interpret the slope and intercept.
March 27, 2015 Today we will be taking an assessment on line of best fit and scatterplots. We will begin solving systems of equations on Monday.
March 26, 2015 Today will be our final day to practice line of best fit and scatterplots. We will be taking an assessment on Friday March 27th.
March 25, 2015 Today we are taking notes on scatter plots and completing a practice assignment. We will be assessing on scatter plots on Friday March 27th.
March 19, 2015 Students will be attending Region 9 today ... no class :(
March 18, 2015
Today we are reviewing how to plot ordered pairs on the coordinate plane, substituting values from an input-output table into a linear equation, and using ordered pairs to graph the equation of a line. These targets are a review from 10th grade Algebra Part 1 and they will help prepare us for our work with scatter plots. | 677.169 | 1 |
Kathleen Becker, PA
I originally bought Algebrator for my wife because she was struggling with her algebra homework. Now only did it help with each problem, it also explained the steps for each. Now my wife uses the program to check her answers. Jason Padrew Sarah Jones, CA
I originally bought Algebrator for my wife because she was struggling with her algebra homework. Now only did it help with each problem, it also explained the steps for each. Now my wife uses the program to check her answers | 677.169 | 1 |
. In particular, it explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay between geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians, as well as for computer scientists and physicists. The book is based on the author's experience in implementing geometric software and includes hundreds of high-quality illustrations.
Facts101 is your complete guide to Fractal Geometry, Mathematical Foundations and ApplicationsIncidence geometry is a central part of modern mathematics that has an impressive tradition. The main topics of incidence geometry are projective and affine geometry and, in more recent times, the theory of buildings and polar spaces. Embedded into the modern view of diagram geometry, projective and affine geometry including the fundamental theorems, polar geometry including the Theorem of Buekenhout-Shult and the classification of quadratic sets are presented in this volume. Incidence geometry is developed along the lines of the fascinating work of Jacques Tits and Francis Buekenhout. The book is a clear and comprehensible introduction into a wonderful piece of mathematics. More than 200 figures make even complicated proofs accessible to the reader.
The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms. This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation. Graduate students, scientists, researchers and practitioners will benefit from this book. The examples given in the text are mostly recent research results, so practitioners can see how to apply geometric algebra to real tasks, while researchers note starting points for future investigations. Students will profit from the detailed introduction to geometric algebra, while the text is supported by the author's visualization software, CLUCalc, freely available online, and a website that includes downloadable exercises, slides and tutorials. provides the foundation for the rest of the material covered throughout; Part Two discusses Euclidean transformations of the plane, as well as groups and their use in studying transformations; and Part Three covers inversive and projective geometry as natural extensions of Euclidean geometry. In addition to featuring real-world applications throughout, Classical Geometry: Euclidean, Transformational, Inversive, and Projective includes: Multiple entertaining and elegant geometry problems at the end of each section for every level of study Fully worked examples with exercises to facilitate comprehension and retention Unique topical coverage, such as the theorems of Ceva and Menalaus and their applications An approach that prepares readers for the art of logical reasoning, modeling, and proofs The book is an excellent textbook for courses in introductory geometry, elementary geometry, modern geometry, and history of mathematics at the undergraduate level for mathematics majors, as well as for engineering and secondary education majors. The book is also ideal for anyone who would like to learn the various applications of elementary geometry.
This monograph-like anthology introduces the concepts and framework of Clifford algebra. It provides a rich source of examples of how to work with this formalism. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work shows that Clifford algebra provides a universal and powerful algebraic framework for an elegant and coherent representation of various problems occurring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics.
Algebraic projective geometry, with its multilinear relations and its embedding into Grassmann-Cayley algebra, has become the basic representation of multiple view geometry, resulting in deep insights into the algebraic structure of geometric relations, as well as in efficient and versatile algorithms for computer vision and image analysis. This book provides a coherent integration of algebraic projective geometry and spatial reasoning under uncertainty with applications in computer vision. Beyond systematically introducing the theoretical foundations from geometry and statistics and clear rules for performing geometric reasoning under uncertainty, the author provides a collection of detailed algorithms. The book addresses researchers and advanced students interested in algebraic projective geometry for image analysis, in statistical representation of objects and transformations, or in generic tools for testing and estimating within the context of geometric multiple-view analysis.
This introduction to modern geometry differs from other books in the field due to its emphasis on applications and its discussion of special relativity as a major example of a non-Euclidean geometry. Additionally, it covers the two important areas of non-Euclidean geometry, spherical geometry and projective geometry, as well as emphasising transformations, and conics and planetary orbits. Much emphasis is placed on applications throughout the book, which motivate the topics, and many additional applications are given in the exercises. It makes an excellent introduction for those who need to know how geometry is used in addition to its formal theory.
"Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and non-Euclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. Regarding its applications, results and techniques of projective geometry are today intensively used in computer vision. This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries." --Publisher's description.
This Handbook deals with the foundations of incidence geometry, in relationship with division rings, rings, algebras, lattices, groups, topology, graphs, logic and its autonomous development from various viewpoints. Projective and affine geometry are covered in various ways. Major classes of rank 2 geometries such as generalized polygons and partial geometries are surveyed extensively. More than half of the book is devoted to buildings at various levels of generality, including a detailed and original introduction to the subject, a broad study of characterizations in terms of points and lines, applications to algebraic groups, extensions to topological geometry, a survey of results on diagram geometries and nearby generalizations such as matroids.
The first geometrical properties of a projective nature were discovered in the third century by Pappus of Alexandria. Filippo Brunelleschi (1404-1472) started investigating the geometry of perspective in 1425. Johannes Kepler (1571-1630) and Gerard Desargues (1591-1661) independently developed the pivotal concept of the "point at infinity." Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made Euclidean geometry, where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-years old Blaise Pascal and helped him formulate Pascal's theorem. The works of Gaspard Monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until Michel Chasles chanced upon a handwritten copy in 1845. Meanwhile, Jean-Victor Poncelet had published the foundational treatise on projective geometry in 1822. Poncelet separated the projective properties of objects in individual class and establishing a relationship between metric and projective properties. The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.
Projective Geometry has been writing in one form or another for most of life. You can find so many inspiration from Projective Geometry also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Projective Geometry book for free.
As an introduction to fundamental geometric concepts and tools needed for solving problems of a geometric nature using a computer, this book fills the gap between standard geometry books, which are primarily theoretical, and applied books on computer graphics, computer vision, or robotics that do not cover the underlying geometric concepts in detail. Gallier offers an introduction to affine, projective, computational, and Euclidean geometry, basics of differential geometry and Lie groups, and explores many of the practical applications of geometry. Some of these include computer vision, efficient communication, error correcting codes, cryptography, motion interpolation, and robot kinematics. This comprehensive text covers most of the geometric background needed for conducting research in computer graphics, geometric modeling, computer vision, and robotics and as such will be of interest to a wide audience including computer scientists, mathematicians, and engineers.
This is one book that can genuinely be said to be straight from the horse's mouth. Written by the originator of the technique, it examines parallel coordinates as the leading methodology for multidimensional visualization. Starting from geometric foundations, this is the first systematic and rigorous exposition of the methodology's mathematical and algorithmic components. It covers, among many others, the visualization of multidimensional lines, minimum distances, planes, hyperplanes, and clusters of "near" planes. The last chapter explains in a non-technical way the methodology's application to visual and automatic data mining. The principles of the latter, along with guidelines, strategies and algorithms are illustrated in detail on real high-dimensional datasets. | 677.169 | 1 |
Mathematics for Chemistry Homework Help
Chemistry is not just a science describing observations of substance transformation. Role of mathematics in chemistry has been grown with the development of physical chemistry, thermodynamics, kinetics and other. Using higher mathematics for solving chemical problems allows to reach the most precious results that can't be received in any different way. Ability to use mathematical apparatus is significant for chemists; he/she must be able to choose correct mathematical method and use it properly. But this requires to know corresponding set of math definitions, methods and algorithms. Most of students have troubles with all this stuff.
Assignment4Student team is ready to solve mathematical chemistry problems you have. We will help you to understand each step, each method; we will answer all questions concerning solution. We are ready to provide additional theoretical material and solutions to additional problems so you could comprehend everything well. Below is the short list describing the categories that our experts are familiar with and can provide effective help with mathematics for chemistry:
Applications of integral calculus (reactions of the first order, reactions of the second and the third order, synchronous and parallel processes, reversible processes, dissolutions, mixed processes, thermokinetics, filtering, ...)
Differential equations solutions (separation of variables, homogeneous equations, total differential equation, linear equations of the first order, equations of the second order, approximation of the solution, quantitative methods) | 677.169 | 1 |
In this appealing and well-written text, Richard Bronson gives readers a substructure for a firm understanding of the abstract concepts of linear algebra and its applications. The author starts with the concrete and computational, and leads the reader to a choice of major applications (Markov chains, least-squares approximation, and solution of differential equations using Jordan normal form).
The first three chapters address the basics: matrices, vector spaces, and linear transformations. The next three cover eigenvalues, Euclidean inner products, and Jordan canonical forms, offering possibilities that can be tailored to the instructor's taste and to the length of the course. Bronson's approach to computation is modern and algorithmic, and his theory is clean and straightforward. Throughout, the views of the theory presented are broad and balanced. Key material is highlighted in the text and summarized at the end of each chapter. The book also includes ample exercises with answers and hints. With its inclusion of all the needed features, this text will be a pleasure for professionals, teachers, and students.
"The text is unusually well and carefully written for a textbook and the exposition is very clear." - George Feldvoss, University of South Alabama
"The quality of the exercises is better than that of Anton. Bronson's exercises seem more original and less trivial. While he does have routine drill problems his non-routine problems require the student to either extend the student's knowledge base or fill in a portion of a proof. The range on these problem sets is quite large, spanning the routine drill problems needed by all students to the theoretical problems for the better students." -Renee Britt - Louisiana State University
"I appreciate the slow increase in the progression of difficulty with proofs... I regard the exposition as superior. Prof. Bronson's text is the best example I've ever seen of motivating definitions in linear algebra, right from the very first page... Bronson incorporates the application first, thus motivating the definition, going from concrete to abstract, instead of the reverse." -Michael Ecker, The Pennsylvania State University
Book Description Academic20887843-2-4 | 677.169 | 1 |
Textbook for Exploring Arithmetic. 170 lessons Grade 3 reviews addition and subtraction skills from Grades 1 and 2 and teaches multiplication and division facts through table 9 and simple two- and three-step multiplication and division. Other concepts drilled are fraction meaning, measure equivalents, money values, and telling time. Mixed computation and Roman numerals are introduced. The pupil does his exercises on paper and the textbook is reusable.
This guide to using the Saxon Homeschool Math and Physics programs has been designed to help homeschoolers using the Math 54 through Calculus and Physics get the most benefit from the program. This is a book full of solutions for the problem book from Saxon Algebra 1
This hardcover textbook has 170 lessons that review and extend skills taught in previous grades. Review in each lesson keeps previous skills current. New concepts include the following areas.
Numbers-Greatest common factor, lowest common multiple, mental shortcuts in calculation.
Fractions-Decimal-fraction equivalents.
Decimals-Multiplying and dividing.
Percents-Discounts, commissions, interest.
Measures-Conversions between English and metric measures, time zones.
Geometry-Perimeter and area of triangles and parallelograms, circumference and area of circles, measuring angles with a protractor.
Graphs-Constructing circle grafts.
The teacher's manual comes in two volumes. Each lesson shows the pupil's pages full size with answers filled in. Extra pages guide the teacher in lesson preparation. Answer keys for quizzes, speed tests, and chapter tests are included | 677.169 | 1 |
books.google.com Teaching of Mathematics in the Elementary and the Secondary School
Popular passages
Page 295 - The four fundamental operations for rational algebraic expressions. Factoring, determination of highest common factor and lowest common multiple by factoring; fractions, including complex fractions, and ratio and proportion; linear equations, both numerical and literal, containing one or more unknown quantities; problems depending on linear equations; radicals, including the extraction of the square root of polynomials and of numbers; exponents, including the fractional and the negative.
Page 77 - And therefore, while in all things that we see or do, we are to desire perfection, and strive for it, we are nevertheless not to set the meaner thing, in its narrow accomplishment, above the nobler thing, in its mighty progress; not to esteem smooth minuteness above shattered majesty ; not to prefer mean victory to honourable defeat; not to lower the level of our aim, that we may the more surely enjoy the complacency of success.
Page 106 - General formulas which men have devised to express groups of details, and which have severally simplified their conceptions by uniting many facts into one fact, they have supposed must simplify the conceptions of a child also.
Page 295 - ... and geometric progressions, with applications. It is assumed that pupils will be required throughout the course to solve numerous problems which involve putting questions into equations. Some of these problems should be chosen from mensuration, from physics...
Page 20 - Give a boy accurate perceptions. Teach him the difference between the similar and the same. Make him call things by their right names. Pardon in him no blunder. Then he will give you solid satisfaction as long as he lives.
Page 127 - With the algebraists, however, who are Pagans themselves, the 'Pagan fables' are believed, and the inferences are made, not so much through lapse of memory as through an unaccountable addling of the brains. In short, I never yet encountered the mere mathematician who...
Page 17 - ... mathematical study furnishes the first scientific key to the existence of bodies and their various motions. Mathematics in its pure form, as arithmetic, algebra, geometry, and the application of the analytical method, as well as mathematics applied to matter and force or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this, its quantitative aspect. Mathematics furnishes the instrument, the tool of thought, which gives...
Page 295 - Quadratic equations, both numerical and literal. Simple cases of equations with one or more unknown quantities, that can be solved by the methods of linear or quadratic equations. Problems depending on quadratic equations. The binomial theorem for positive integral exponents. The formulas for the nth term and the sum of the terms of arithmetical and geometric progressions, with applications.
Page 60 - ... that is to say of eight feet ; and I want to know whether you still say that a double square comes from a double line? BOY. Yes. Soc. But does not this line become doubled if we add another such line here? BOY. Certainly. Soc. And four such lines will make a space containing eight feet? BOY. Yes. Soc. Let us describe such a figure: is not that what you would say is the figure of eight feet? | 677.169 | 1 |
Structured programming with high-level language using FORTRAN. Control structures. Different data types with emphasis on real and complex number computations. Arrays used with vector and matrix manipulation to solve simultaneous equations. External subroutines for mathematical and graphical applications. Prerequisite or corequisite: Math 1220 or MATH 1231. | 677.169 | 1 |
This book contains 40 lessons in problem solving divided into five chapters. Chapter 1 contains several lessons in which solutions are derived through investigation of numerical patterns. Chapter 2 contains lessons in which problem-solving situations are presented in a graphical context. It also includes two lessons in which students construct their own graphs, then use those graphs to solve problems. Chapter 3 contains lessons in which information is presented in a picture or chart format, and students use this information to solve related problems. The lessons in this chapter involve such things as telephone and electric bills, time zones, and tournaments. Chapter 4 involves problem solving in a geometric context and contains lessons involving both two-dimensional and three-dimensional geometry. Lessons in Chapter 5 involve problem solving in a probability context. (JRH) | 677.169 | 1 |
Courses
To see the schedule of classes, go to Schedule of Classes (within Ask Banner), select "MATH Mathematics" in the "Department Menu," and choose the semester you are interested in for a complete listing of courses offered.
Mathematics and Statistics: I. Introductory
121a. Single Variable Calculus 1
The calculus of one variable and its applications are discussed. Topics include: limits, continuity, derivatives, applications of derivatives, transcendental functions, the definite integral, applications of definite integrals, approximation methods, differential equations, sequences, and series. The department.
Prerequisite(s): a minimum of three years of high school mathematics, preferably including trigonometry.
Mathematics 121 is not open to students with AP credit in mathematics or students who have completed MATH 101 or its equivalent.
Yearlong course sequence 121, MATH 126/MATH 127.
126a and b. Calculus IIA: Integration Theory 0.5
In this course, we expand and build upon basic knowledge of differential and integral calculus. Various techniques and applications of integration will be studied. The calculus of transcendental functions---such as the exponential, logarithmic, and inverse trigonometric functions---will also be developed. A main theme in this course is the many ways functions can be defined, and arise naturally in problems in the mathematical sciences.
Prerequisite(s): MATH 121 or its equivalent.
First or second six-week course.
127a and b. Calculus IIB: Sequences and Series 0.5
Real numbers may be represented as infinite decimals. In this course we generalize this representation by studying the convergence of sequences and of series of real numbers. These notions further generalize to the convergence of sequences and series of functions. We study these ideas and their relation to the Calculus.
Prerequisite(s): MATH 121 or its equivalent.
First or second six-week course.
131 Numbers, Shape, Chance, and Change 1
What is the stuff of mathematics? What do mathematicians do? Fundamental concepts from arithmetic, geometry, probability, and the calculus are explored, emphasizing the relations among these diverse areas, their internal logic, their beauty, and how they come together to form a unified discipline. As a counterpoint, we also discuss the "unreasonable effectiveness" of mathematics in describing a stunning range of phenomena from the natural and social worlds. The department.
Prerequisite(s): at least three years of high school mathematics.
Open only to freshmen; satisfies the college requirement for a Freshman Writing Seminar.
Two 50-minute periods and one 50-minute discussion period.
132 Mathematics and Narrative 1
To most, mathematics and narrative live in opposition-narrative is ubiquitous while mathematics is perceived as inscrutably esoteric and obscure. In fact, narrative is a fundamental part of mathematics. Mathematical proofs, problems and solutions, textbooks, and journal articles tell some sort of story. Conversely, many literary works (Arcadia, Proof, and Uncle Petros and the Goldbach Conjecture) use mathematics as an integral part of their narrative. Movie and television narratives such as Good Will Hunting and Numb3rs are also mathematically based. Nonfiction works about mathematics and mathematical biographies like Chaos, Fermat's Enigma, and A Beautiful Mind provide further examples of the connection between mathematics and narrative. We use this course to explore this connection by reading and writing a variety of mathematical narratives.
Open only to freshmen; satisfies college requirement for a Freshman Writing Seminar.
Not offered in 2017/18.
141 Introduction to Statistical Reasoning 1
The purpose of this course is to develop an appreciation and understanding of the exploration and interpretation of data. Topics include exploratory data analysis, basic probability, design of studies, and inferential methods including confidence interval estimaation and hypothesis testing. Applications and examples are drawn from a wide variety of disciplines. When cross-listed with biology, examples are drawn primarily from biology. Statistical software is used. Computationally less intensive than MATH 240. Ming-Wen An, Jingchen Hu.
Prerequisite: three years of high school mathematics.
Not open to students with AP credit in statistics or students who have completed MATH 240, ECON 209 or PSYC 200.
Not recommended for students who have taken a semester of calculus: those students should instead consider MATH 240. AP Statistics, MATH 141 and MATH 240 all provide an introduction to statistics and students should not take more than one; they all can serve as a prerequisite for further statistics courses in the Mathematics and Statistics Department.
In certain semesters, one section may be cross-listed with BIOL 141.
Three 50-minute periods.
142a. Statistical Sleuthing: Personal and Public Policy Decision-Making in a World of Numbers 1
The world inundates us with numbers and pictures intended to persuade us towards certain beliefs about our health, public policy, or even which brand of product to buy. How can we make informed decisions in this context? The goal of this course is for us to become statistical sleuths who critically read and summarize a piece of statistical evidence. We read articles from a variety of sources, while using basic statistical principles to guide us. Course format: mixture of discussion and lecture, with regular reading and writing assignments. The department.
Open only to freshmen; satisfies the college requirement for a Freshman Writing Seminar.
221a and b. Linear Algebra 1
Prerequisite(s): MATH 126 and MATH 127 or equivalent, or permission of the department.
228b. Methods of Applied Mathematics 1
Survey of techniques used in the physical sciences. Topics include: ordinary and partial differential equations, series representation of functions, integral transforms, Fourier series and integrals. The department.
Prerequisite(s): MATH 126 and MATH 127, or permission of the department.
240 Introduction to Statistics 1
The purpose of this course is to introduce the methods by which we extract information from data. Topics are similar to those in MATH 141, with more coverage of probability and more intense computational and computer work. Ming-Wen An, Jingchen Hu.
Prerequisite(s): MATH 126 and MATH 127, or permission of the department.
242a. Applied Statistical Modeling 1
Applied Statistical Modeling is offered as a second course in statistics in which we present a set of case studies and introduce appropriate statistical modeling techniques for each. Topics may include: multiple linear regression, logistic regression, log-linear regression, survival analysis, an introduction to Bayesian modeling, and modeling via simulation. Other topics may be substituted for these or added as time allows. Students will be expected to conduct data analyses in R. The department.
Prerequisite(s): MATH 126 and MATH 127, or permission of the department.
268 Protecting Information: Applications of Abstract Algebra 1
In today's information age, it is vital to secure messages against eavesdropping or corruption by noise. Our study begins by surveying some historical techniques and proceeds to examining some of the most important codes currently being used to protect information. These include various public key cryptographic schemes (RSA and its variants) that are used to safeguard sensitive internet communications, as well as linear codes, mathematically elegant and computationally practical means of correcting transmissions errors. The department.
Prerequisite(s): MATH 126 and MATH 127, or permission of the department.
Not offered in 2017/18.
290a or b. Field Work 0.5 to 1
297a. Topics in Mathematics 0.5
Reading Course
Prerequisite(s): MATH 221 or equivalent, and permission of the instructor.
298a or b. Independent Work 0.5 to 1
Election should be made in consultation with a department adviser.
Mathematics and Statistics: III. Advanced
301b. Senior Seminar 0.5 to 1
Areas of study and units of credit vary from year to year. The department.
Open only to seniors who have a declared major in mathematics. It is strongly recommended that MATH 361 be completed before enrolling in Mathematics 301.
315 Advanced Topics in Applied Mathematics 1
This course introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. We develop such equations from real-world examples and applications from physics, biology, epidemiology, and from student interests. We explore several mathematical and numerical strategies for solving these models and introduce ways to glean information from them. Students may also be exposed to a collection of non-standard mathematical modeling techniques: Cellular Automata, Pair Approximation Equations, and Agent Based Modeling. Familiarity with programming in Matlab is developed throughout the semester.
Prerequisite(s): MATH 220, MATH 221 and MATH 228.
321a. Real Analysis 1
A rigorous treatment of topics in the classical theory of functions of a real variable from the point of view of metric space topology including limits, continuity, sequences and series of functions, and the Riemann-Stieltjes integral. The department.
324b. Complex Analysis 1
Integration and differentiation in the complex plane. Topics include: holomorphic (differentiable) functions, power series as holomorphic functions, Taylor and Laurent series, singularities and residues, complex integration and, in particular, Cauchy's Theorem and its consequences. The department.
341 Statistical Inference 1
An introduction to statistical theory through the mathematical development of topics including resampling methods, sampling distributions, likelihood, interval and point estimation, and introduction to statistical inferential methods. The department.
Prerequisite(s): MATH 220, MATH 221 and MATH 241.
Three 50-minute periods.
342a. Applied Statistical Modeling 1
For students who have completed MATH 341. Students in this course attend the same lectures as those in MATH 242, but will be required to complete extra reading and problems. Ming-Wen An.
351 Mathematical Logic 1
An introduction to mathematical logic. Topics are drawn from computability theory, model theory, and set theory. Mathematical and philosophical implications also are discussed. The department.
Prerequisite(s): MATH 321 or MATH 361.
Not offered in 2017/18.
361b. Modern Algebra 1
The theory of groups and an introduction to ring theory. Topics in group theory include: isomorphism theorems, generators and relations, group actions, Sylow theorems, fundamental theorem of finite abelian groups. The department. | 677.169 | 1 |
Patterns are a ubiquitous phenomenon. They exist in nature's plant species emerging as a complete order from truncated matter. Patterns are also present in the fine arts where the aesthetic properties of form exist and conjugate through paintings and sculptures. Mathematical patterns, the focus of this position paper, dominate financial scenarios, growth and decay problems, as well as population rates. These topics can be articulated through the language of recursion where a present numerical value occurs in the sequential context of prior ones. Many college students have little or no knowledge of this recursion concept, and traditional analytic pedagogy is often characterized by tedium. The TI-84's designated sequence mode is modeled and implemented here as a paragon for students to define, investigate and visually perceive recursively defined problem solving tasks in terms of the natural value, n. The sequence mode structurally delineates the recursion function by translating present and antecedent iterations associated with the following conceptions: (1) a computational sequence; (2) a financial scenario; and (3) a medication problem. These three tasks are conveyed by applying n as an independent variable, a table of integer or discrete outcome values, and a graphic representation of a disconnected set of points. The TI-84 sequence graphing mode edifies the cognition of advanced algebraic topics for students. | 677.169 | 1 |
GUIDELINES ON THE USE OF ELECTRONIC CALCULATORS IN NATIONAL EXAMINATIONS 1. The scientific calculators used in the PSLE Mathematics and Foundation Mathematics examinations should have the following minimum features: (a) four arithmetic operations (, , , ) including the order of operations and use of brackets; (b) fractional function which retains the fraction in the form 'a/b' and 'a b/c'; (c) square root ( ) and cube root ( 3 ) functions; and (d) value of Pi ().
2. The following guidelines are to be adhered to in the use of scientific calculators in national examinations. (a) The calculator must be silent, with a visual display only. (b) The working condition of the calculator is the responsibility of the pupil and a fault in a calculator cannot be used as a reason for seeking special consideration for the user. (c) Calculators must not be borrowed from other pupils in the course of the examination for any reason. (d) External storage media, e.g. card, tape, and plug-in modules, must not be in the possession of pupils during the examination. (e) No unauthorised materials, e.g. instructions leaflets, formulae printed on the lid or cover of a calculator or similar materials, must be in the possession of pupils during the examination (where the instructions cannot be removed they should be securely covered). (f) No programmable calculators are allowed. (g) No calculators with permanent features of a programmed kind are allowed, e.g. calculators capable of numerical integration, numerical differentiation, and/or expressing in irrational number form. (h) No calculators with special communication features are allowed, e.g. calculators with the capability of remote communication via infra-red or blue-tooth with other machines. (i) No calculators with capabilities for storing and displaying visual and verbal information are allowed. 3. Graphic calculators are allowed for use ONLY in subjects examined at H1, H2 and H3 Levels of the GCE A-Level Curriculum. Candidates are to note the following: (a) No prepared programs may be taken into the examination room. Information (including text or formulae) and/or programs stored in the calculator's memory must be cleared before the examination. (b) No calculators with built-in symbolic algebra manipulations, symbolic differentiation or integration are allowed. 4. The use of any calculator with unacceptable features or for retrieval of information/programs during the examinations is an infringement of the regulations. 5. The list of approved models of scientific and graphic calculators can be found in Annex A. | 677.169 | 1 |
Mathematics GCSE
Description
The topics covered are statistics, number, space and shape and Algebra. There are three modules as follows:-Module l: Statistics (studied at Higher tier) Module 2: Number (studied at Higher tier) Module 3: Space and shape and algebra (studied at Foundation tier). Modules l, 2 and 3 are assessed by examinations. These would normally be held in November, March and June respectively. There is one paper for each examination. There will be weekly assignments set and marked by the tutor.
Teaching
Staff are highly qualified and enthusiastic. They ensure learner success by making the teaching and learning differentiated and learner centred. Teaching styles vary from lecture to group work, to ICT based research to practical classroom activities. Tasks are set to support and reinforce the subject material covered in the sessions. It is expected that these tasks are completed in the learner's own time.
Entry Requirements
Applicants must hold an existing minimum grade D or grade 3 at GCSE. Functional Skills Maths at Level 2 and/or Adult Numeracy Level 2 qualification achieved within the last three years will also be considered. For all learners there will be stringent screening requirements which will take place at interview and/or enrolment. This usually involves an aptitude test and an on line initial assessment
Interview Process
An interview with specialist tutor
Questions & Answers
How is the course assessed?
Assessment is by examination which takes place in June. There will be weekly assignments set and marked by the tutor.
Are there any opportunities for progression?
To any level 3 programme of study. To employment - many employers require a pass at GCSE Maths. To some university degree courses together with relevant level qualifications, particularly teaching, nursing and psychology | 677.169 | 1 |
Calculator"?
A calculator is a tool for finding answers to mathematical problems. Smartphones almost universally carry a calculator and many third party options are available in app form, with most versions typically offering robust features that would make them scientific calculators.
See a typo or something that needs a correction or clarification? Send us feedback and let us know. | 677.169 | 1 |
The class is challenging, but it shows where and why we use Numerical Linear Algebra in so much of our lives: from iterations and eigenvalues, to singular value decomposition and pixels, you'll learn so much in this class!
Course highlights:
I learned different types of methods in computation and why linear algebra was important. I think this class should be taken by every math major.
Hours per week:
3-5 hours
Advice for students:
You should know a little bit of computer programming coming into this course as you'll be required to code algorithms in MATLAB. | 677.169 | 1 |
Math 1000
Math 1000 Course Description
Math 1000 is an introductory level algebra course designed to prepare students to be successful in Math 1310.
This course is worth two credit hours and will not satisfy the Math General Education requirement.
Successful completion of this course will satisfy the prerequisite for Math 1310, Intermediate Algebra.
Course Design Overview
Class meets once a week for two hours and 15 minutes at the scheduled time. In class, your teacher will cover important concepts, work especially difficult problems, and guide you through the work that will be done each week. Class time will be separated into the first hour of lecture and the second hour for taking computer-assisted homework, quizzes and tests.
Your teacher will discuss study strategies and help you to avoid common errors.
Topics Covered
Whole numbers, integers, fractions and mixed numbers, decimals.
Simplifying mathematical expressions, the properties of equality.
Solving linear equations in one variable and using linear equations to solve problems.
Ratios and proportions, graphing and the rectangular coordinate system | 677.169 | 1 |
26 Used
About this product
Description
Exam Board: AQALevel & Subject: GCSE MathsFirst teaching: September 2015 First exams: June 2017This Maths all-in-one revision and practice book is guaranteed to help you achieve the best results. Containing clear and accessible explanations of all the GCSE content, there are lots of practice opportunities for each topic throughout the book. | 677.169 | 1 |
9780471669609
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Summary
This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation. Clear, accessible, step-by-step explanations make the material crystal clear. The authors spotlight the relationships between concepts to give a unified and complete picture. Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues. | 677.169 | 1 |
Want to excel in your basic math courses? Look no further than this enhanced 3-panel "Quizzers" edition of our best-selling Math Fundamentals guide. A comprehensive list of sample problems—and their answers for easy checking of work—newly included with the guide provide better understanding of such math basics as decimals, integers, ratios, and more. The same great tips, examples, and diagrams from the original version can also be found here | 677.169 | 1 |
Author: JJtheTutor | Isbn: 9781508732730 | File size: 72.34MB | Year: 2015 | Pages: 96 | Language: English | File format: PDF | Category: Mathematics What every student should know and master prior to starting his or her first College level Calculus course. This book is designed to help a student that is preparing for a Calculus course. The Prep-Course book is an isolation of everything that is crucial from previous...
Author: Jonathan Tullis | Isbn: 9781532836350 | File size: 105.85MB | Year: 2016 | Pages: 32 | Language: English | File format: PDF | Category: Mathematics All of the most common formulas, identities and theorems needed from Algebra with examples, tips and tricks. The purchase of this book includes access to WeSolveThem.com based on the current promotion being offered. Check WeSolveThem.com for current promotion. Download...
Author: Paul Fannon | Isbn: 9781107613065 | File size: 39MB | Year: 2012 | Pages: 679 | Language: English | File format: PDF | Category: Mathematics This title forms part of the completely new Mathematics for the IB Diploma series. This highly illustrated coursebook, available in both print and e-book formats, has been written to specifically cover the new IB Standard Level syllabus. Based on the new group 5 aims, the...
Author: Jonathan Tullis | Isbn: 9781515177128 | File size: 72.37MB | Year: 2015 | Pages: 96 | Language: English | File format: PDF | Category: Mathematics After years of tutoring students from various schools covering many subjects, the information in this text is what I found to be the most important and forgotten from previous courses. This book is designed to be a reference guide for students that need a refresher of...
Author: Limsoon Wong | Isbn: 1580532918 | File size: 8MB | Year: 2004 | Pages: 176 | Language: English | File format: PDF | Category: Mathematics The Institute for Mathematical Sciences at the National University of Singapore organized a program on "Post-Genome Knowledge Discovery" from January to June 2002. The program focused on the computational and statistical analysis of sequences and genetics, and the mathematical...
Author: JJtheTutor | Isbn: 9781508750543 | File size: 37.89MB | Year: 2015 | Pages: 48 | Language: English | File format: PDF | Category: Mathematics What every student should know and master prior to starting his or her first College level Algebra course. This book is designed to help a student that is preparing for a college algebra course. The Prep-Course book is an isolation of everything that is crucial from previous...
Author: R. Remmert, Siegfried Bosch, and U. Güntzer | Isbn: 3540125469 | File size: 15.4MB | Year: 1984 | Pages: 452 | Language: English | File format: PDF | Category: Mathematics In the book of BGR (= Bosch-Guntzer-Remmert) a systematic approach to Tate's theory is provided in 415 pages. The book was planned in the late sixties and drafts of a large part of it existed by 1970. It consists of a long part on valuation...
Author: Foster N. | Isbn: 9781540314642 | File size: 7.79MB | Year: 2016 | Pages: 346 | Language: English | File format: PDF | Category: Mathematics This book describe the different data and image types, and give details about how to read, write, work with, and display graphics images; how to alter the display properties and aspect ratio of an image during display; how to print an image; and how to convert the data type...
Author: James Tanton | Isbn: 9780883858363 | File size: 6MB | Year: 2016 | Pages: 232 | Language: English | File format: PDF | Category: Mathematics This guide covers the story of trigonometry. It is a swift overview, but it is complete in the context of the content discussed in beginning and advanced high-school courses. The purpose of these notes is to supplement and put into perspective the material of any course on... | 677.169 | 1 |
Does somebody here know anything about printable positive and negative integers number lines for students ? I'm a little confused and I don't know how to answer my algebra project regarding this topic. I tried reading all materials about it that could help me figure things out but I still can't finish it. I'm having a hard time answering it especially the topics inverse matrices, simplifying expressions and graphing equations. It will take me days to answer my algebra homework if I can't get any assistance. It would really help me if somebody can suggest anything that can help me with my algebra homework.
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trigonometry, rational inequalities and angle complements were a nightmare for me until I found Algebra Helper , which is truly the best math program that I have ever come across. I have used it frequently through several math classes – Algebra 1, Algebra 2 and Intermediate algebra. Just typing in the algebra problem and clicking on Solve, Algebra Helper generates step-by-step solution to the problem, and my math | 677.169 | 1 |
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Oh want to sayI,ve found that the science section has physics included in the test, I,m not all that kean to physics,but more with earth science. These sites make some of the most dependable versions of the software but should always be checked before anything continues with it. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, and language. Note: We answer to algebra word problems automoderator for a few things. The others, such as TIDAL, Grooveshark, Rdio, and …GitHub logins are hacked with stolen passwords, Facebook Messenger gets a hidden football game, Snapchat, Scale factor volume and Twitter evolve their advertising, Now On Tap gets a useful upgrade, and NASA conducts a fire barclays online banking change address roles Take review and enrichment to mathematics word search next level. Back to top SanG Registered: 31. 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A resource for students, teachers and parentsFIGURE THIS. They are both directly aligned with each other and work together to help reinforce and assess math concepts all year. Click on "Find Your Book: Go"5. WORLD RECORD: The largest American flag ever made was first displayed on March 22,1980, and measures 411 by 210 timed addition and subtraction worksheets. Large collection of exams sorted by topics. Algebraic equations used in every day life. ConsThis is an educational tool, not for scientific calculations. The problem with the CPM program is not evident until CPM students enter the traditional Pre-Calculus level. For instance, abstract algebra arises by relaxing the axioms of a vector space. What is social policy and how is it related to other public policy. Hell, freaking VisualBasic had a form of type inference and JIT a decade ago. Reply HELP for help, STOP to cancel. Most of the time you can receive a detailed list including all the steps for solving scale factor volume math homework problem. That the location of the direct instruction books by Larson, Boswell, Kanold and Stiff be disclosed immediately and be made available so that they may continue to be used as the primary resource for mathematics instruction in 4J, and that the CPM materials already purchased be used as a supplement solve fractions calculator these direct instruction books. Text: Applied Linear Algebra by Olver and Shakiban. Guidance for providers of SCaSP. At school I was running FF 3.
This did scale factor volume mesh with the scale factor volume from Pennsylvania, Wisconsin, Utah, or New York, where Common Core is alleged to lower standards - in one case, specifically, to move multiplication tables from third grade to fifth grade. Her teacher said that she cannot give her one-on-one tutoring. Draw a scatter plot for the data. Algebra 2 Workbook No Answers. The Student Success Center's Math Lab is located in MC 3. We also realize that there are tons of things competing for your time other classes, work, kids, friends, family, social events…basically, life. Seriously overusing smileys and color and fonts will make you come off like a giggly teenage girl, which is not generally a good idea unless you are more interested in sex than answers. We welcome all new members and hope to see you around a lot. Students shall be awarded one-half to one credit for successful completion of this course. However, there are many educational math games that teach important concepts while also keeping math calendar worksheets entertained. | 677.169 | 1 |
Math Department
The goals of the Mathematics Department are to improve and extend arithmetical skills, to develop a facility with the kinds of problems most citizens meet, to develop a facility with mathematical skills and concepts necessary to use in jobs, technical schools, armed services, and college work; and, to provide personal enjoyment. It is the goal of the department to provide opportunities for each student to progress in mathematics as far as personal ability, maturity, interest, and time allow.
mnb
6 credits are required for graduation
Required Courses
Students will take courses which best meet their needs, as determined by level and success in previous coursework. Math is sequential and classes must be completed successfully before moving on to the next course. Regardless of the first math class taken, all students must earn 3 years of math (6 credits) in order to graduate. This may range from Ramp Up to Algebra, Algebra 1-2, and Geometry to Honors Algebra 3-4, Honors Trigonometry/Pre- Calculus, and AP Calculus. It all depends on the individual student's point of entry in the sequence. | 677.169 | 1 |
exploration of tools and mathematics and issues in mathematics education related to tool use. The book has four parts. The first part sets the scene with a reflection on doing a mathematical task with different tools, a mathematician's account of tool use in his work and historical considerations of tool use. The second part opens with a broad review of technology and intellectual trends, circa 1970, and continues with three case studies of approaches in mathematics education and the place of tools in these approaches. The third part considers issues related to mathematics instructions: curriculum, assessment and policy; the calculator debate; mathematics in the real world; and teachers' use of technology. The final part looks to the future and digital tools: task design; the importance of artefacts in gameplay; and new forms of activity via connectivity.Read more...
Abstract:
This book offers an exploration of tools and mathematics and issues in mathematics education related to tool use, from pre-history to future directions in the field. It includes coverage of curriculum, assessment, and policy design.Read more... | 677.169 | 1 |
Algebra I (Course M145) and II (Course M155)
Landmark's Freedom Baptist Curriculum
01 March 2016
Landmark's Freedom Baptist Curriculum Algebra I course is strongly academic but still suitable for independent study. It teaches math theory and vocabulary, as well as applications and skills more comprehensively than we find in Saxon. It lays a solid foundation necessary for higher math studies. It requires more work and study than some other programs, but clear explanations enable students to do it on their own, especially with the help of the optional solutions manual.
It reviews basic arithmetic skills quickly at the beginning then proceeds to cover all basic Algebra I course material up through fairly extensive coverage of quadratic equations. It introduces functions and graphing of quadratic equations at the end, topics usually reserved for Algebra II.
The presentation lacks polish; you can easily spot where some revisions have been made to the course. While there are some word problems, it is a very theory-based approach with no hands-on work and little application practice. The course includes two student worktexts (called study guides), answer keys, weekly quizzes and answers, and quarterly tests and answer keys.
Algebra II is similar in format to Algebra I. However, I am concerned that it might be too difficult for many students without a teacher present because it covers some challenging topics too quickly. Some topics for which coverage seems too brief are slope, graphing, parabolas, and determinants.
Pricing Information
All prices are provided for comparison only and are subject to change. Click on prices to verify their accuracy. | 677.169 | 1 |
Hornsby/Lial/Rockswold's Graphical Approach covers functions through a consistent four part analytical process that asks students to 1) Examine the nature of the graph 2) Solve a typical equation analytically and graphically 3) Solve the related inequality analytically and graphically, and finally, 4) Apply analytic and graphical methods to solve an application of that class of function0321927338 / 9780321927330 A Graphical Approach to Algebra and Trigonometry
"synopsis" may belong to another edition of this title.
About the Author:
John Hornsby: When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics, education, or journalism. His ultimate decision was to become a teacher, but after twenty-five years of teaching at the high school and university levels and fifteen years of writing mathematics textbooks, all three of his goals have been realized; his love for teaching and for mathematics is evident in his passion for working with students and fellow teachers as well. His specific professional interests are recreational mathematics, mathematics history, and incorporating graphing calculators into the curriculum. John's personal life is busy as he devotes time to his family (wife Gwen, and sons Chris, Jack, and Josh). He has been a rabid baseball fan all of his life. John's other hobbies include numismatics (the study of coins) and record collecting. He loves the music of the 1960s and has an extensive collection of the recorded works of Frankie Valli and the Four Seasons.
Marge Lial (late) was always interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, was most recently affiliated with American River College. An avid reader and traveler, her travel experiences often found their way into her books as applications, exercise sets, and feature sets. She was particularly interested in archeology, and trips to various digs and ruin sites produced fascinating problems for her textbooks, involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan.
Gary Rockswold has been teaching mathematics for 33 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate, and graduate students, and adult education classes. Now retired, he most recently served as professor of mathematics at Minnesota State University–Mankato. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his lovely wife and two children. | 677.169 | 1 |
Product Description
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Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures! Each lesson in Saxon Math's textbook is taught step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; and because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. After the lesson, students complete the 30-question Problem Set in the Saxon text; a few problems in the set come from the new concept, however most review previous lessons. Each problem is cross-referenced back to the lesson where it was taught, allowing students to quickly review any concept.Taught from a Christian worldview, Dr. David Shormann also provides a weekly syllabus to help students stay on track with the lessons. DIVE teaches the same concepts as Saxon, but does not use the problems given in the text; it cannot be used as a solutions guide.
Algebra 2 covers traditional second year algebra topics as well as a semester of geometry, real world problems, linear and nonlinear equations, statistics and probability, graphing and basic trigonometry. This DIVE CD can be used with Saxon Algebra 2's 2nd and 3rd Editions; the CLEP Professor College Algebra CLEP Exam prep course is also included.
System Requirements:
Mac OS 10.3.9-10.4
Windows 10,8,7,98, 2000, ME, XP, Vista, 8, 10
Quicktime Download Required
IMPORTANT! PLEASE NOTE: This CD plays in VLC instead of Flash Player. Please double-click the Open Me First folder and follow the installation instructions to have your CD play correctly.
The instructor does a fine job in explaining each lesson.After explaining the new lesson,he provides 2-3 sample problems. He then encourages the student to pause and try the problem first and then check their work to his solution.
We decided on Saxon Math because this was what was recommended to us, as a homeschooling family. We find it useful because the books and materials are well written, and the CD Rom really helpful as a teaching tool. We are weak in Algebra but the CD Instructor is very clear. And this has helped with our son's work (currently in Grace 10) for the last 3 years. In fact, the CD Rom is so well made that he self studies with it and the books and materials given to him | 677.169 | 1 |
Solutions for math problems
The calculus page problems list problems and solutions developed by : d a kouba and brought to you by : ecalculusorg. Webmathcosmeocom webmathcosmeocom. Webmath is designed to help you solve your math problems composed of forms to fill-in and then returns analysis of a problem and, when possible, provides a step-by. Mixture problems with solutions more math problems with detailed solutions in this site popular pages math problems, questions and online self tests. This book contains a selection of more than 500 mathematical problems and their solutions from the phd qualifying examination papers of more than ten famous american.
Math practice problems for 1st grade counting one-digit addition one-digit subtraction problems for 2nd grade. Solve calculus and algebra problems online with cymath math problem solver with steps to show your work get the cymath math solving app on your smartphone. Solution to problem 1: the circumference of a circle is given by c = 2 pi r, where r is the radius of the circle substitute c by 72 pi to obtain the equation. Math homework help hotmath explains math textbook homework problems with step-by-step math answers for algebra, geometry, and calculus online tutoring available for.
Sample problems are under the links in the sample problems column and the corresponding review material is under the concepts column new problems are given each. Need urgent help with math problems no panic we are here to cope with your complex math assignment exceptional quality is guaranteed. Most of the classes have practice problems with solutions available on the practice problems pages common math errors algebra (practice problems.
Sometimes gmat students, especially folks who are not very confident in math, think about gmat math problems in a very binary framework — they get the problem.
Free math lessons and math homework help from basic math to algebra, geometry and beyond students, teachers, parents, and everyone can find solutions to their math. A collection of problems in di erential calculus problems given at the math 151 solution to the detailed solution o ered by the textbook. | 677.169 | 1 |
Clothes Washers life cycle - Cost and environmental performance
the cross-disciplinary approach used by two groups of students from Everett Community College working together in order to numerically evaluate and compare the cost and environmental performance of different clothes washers in the market. The two classes involved were Intermediate Algebra and Green Purchasing.
Curriculum designed
to develop students quantitative skills college-wide ability.
Saddle-Making
This module uses the example of saddlemaking to illustrate practical uses of simple arithmetic in a small business setting. Using proportions and percentages to make business decisions as a professional saddlemaker, this unit analyzes materials and production costs as well as pricing simple and complicated work. To see this entire unit as a module on The Electronic Bookshelf, click here. | 677.169 | 1 |
I've always wanted to learn download basic maths formule, it seems like there's a lot that can be done with it that I can't do otherwise. I've searched the internet for some useful learning resources, and consulted the local library for some books, but all the information seems to be directed to people who already know the subject. Is there any resource that can help new people as well?
It always feels good when I hear that students are willing to put that extra effort into their studies. download basic maths formule is not a very complex subject and you can easily do some initial work yourself. As a useful tool, I would suggest that you get a copy of Algebrator. This software is quite beneficial when doing math yourself. | 677.169 | 1 |
Year 12 Mathematics 3CD
Overview
In this course students will develope their knowledge of calculus concepts (differentiation and integration) and their algebraic, graphing and calculus skills, and apply these in mathematical modelling. They will use counting techniques and probability laws, and calculate and interpret probabilities for the binomial, uniform and normal random variables. Students will model data with probability functions and analyse data from samples. They will justify decisions and critically assess claims about data. They will solve systems of equations in three variables and linear programming problems. They will verify and develop deductive proofs in algebra and geometry. They will use mental and written methods and technologies where appropriate.
Topics
estimation and calculation
functions, graphs and patterns
equations and inequalities
finance
rate
networks
quantify and interpret chance
collect, organize, represent and interpret data
measurement
conduct chance experiments
quantify and interpret chance
calculus
reason geometrically and algebraically
Further elaboration of these topics may be found in program of learning issued to the students at the commencement of the course.
A brief interim report on class work and behaviour will be issued towards the end of Term 1. The achievement for the course will be reported as a College A - E grade as well as a percentage mark for the course and an examination mark for both semester 1 and the year. In addition, the student's approach to class work and behaviour will be reported via the usual key performance indicators.
Student Expectations
Students are expected to:
bring the required text, calculator and stationery to each lesson
complete regular homework from a variety of sources including the text, worksheets and MathsOnline
be aware of assessment dates and have prepared thoroughly for each assessment | 677.169 | 1 |
You'll gain access to interventions, extensions, task implementation guides, and more for this lesson plan.
Lesson objective: Identify nonlinear functions based on tables and graphs.
This lesson helps to build fluency with finding rates of change to determine linearity. Tables and graphs are used here because those are two of the representations with which students need to become fluent. This work develops students' understanding that not all functions are linear.
Students engage in Mathematical Practice MP.5 (use appropriate tools strategically) as they decide whether to use a table or a graph to investigate rates of change. Is one representation easier to analyze than another? A point on graphs may be harder to read than in tables, so students should think about which representation gives a mathematically precise answer. | 677.169 | 1 |
The Foundation Program Department of Mathematics (FPDM) offers two courses, Elementary Algebra (Basics of Mathematics, Solving different types of equations and inequalities) and Pre-Calculus (Different types of Functions, their properties and Trigonometry). The objective of the program is to provide students with a solid foundation in mathematical skills in preparation for their subsequent entry into academic programs at Qatar University. Through the use of various e-learning course management tools and an interactive learning environment, FPDM offers and maintains an innovative learning and teaching environment which develops students' critical thinking skills, promotes independent as well as lifelong learning capabilities, and equips students with the mathematical skills required for productive participation in society | 677.169 | 1 |
Mathematical Applications for Agriculture(Spiral spine)
Synopsis
Many colleges and universities have discovered that students enrolled in their agriculture programs are lacking in their abilities to perform mathematically and have varying skill levels. Additionally, many of these students are unable to apply previously learned mathematical skills to real-world, agriculturally significant situations. This user-friendly text/reference is designed to:* provide the student with a solid foundation in practical mathematical applications to agriculture* provide the student with the opportunity to develop his/her thinking skills by the way the problems are designed* provide the student with some real world agriculturally- applicable facts and situations. All problems in the book are application problems. Part I deals with mathematical operations involving fractions, area, volume, slope, graphs, etc. It contains three chapters and is the foundation for the rest of the book. Part II deals with mathematics involved in crop production such as soil preparation, harvesting, storage, etc. for such crops as corn, wheat, soybeans, cotton and more. Part III deals with the mathematics involved in livestock production such as cattle, swine, poultry, and more.
Part IV deals with the mathematics involved in agribusiness such as elevators/milling companies, fertilizer/chemical suppliers, feed suppliers, etc. and mathematics involved in agribusiness and personal finances. A handy Appendix contains a quick review of arithmetic, applied geometry, algebra, as well as tables of measurements. Ag Facts and Chapter Skill Tests are available to maintain the students' interest and monitor skill progression. A helpful Teacher's Solutions Manual is also available | 677.169 | 1 |
Breadcrumbs
BSc - MathematicsMathematics is a subject of vital importance that underpins many activities of our modern world. Employers recognise University of Waikato Mathematics graduates for their analytical and problem solving skills as well as their high level of numeracy, which are critical skills for a broad range of jobsData Analyst
Financial Analyst
IT or Computing Analyst
Mathematical Modeller
Meteorologist
Research Scientist
Secondary School Teacher
Papers
Papers available within this subject
Mathematics is a language of fundamental importance which underpins many activities of society. It plays a crucial role, both theoretically and practically, in many areas such as science, computing, economics and finance.
Mathematics is available as a major for the Bachelor of Computing and Mathematical Sciences with Honours and the Bachelor of Science. Mathematics may also be included as a second major or minor in other undergraduate degrees, subject to the approval of the Faculty in which the student is enrolled.
To complete Mathematics as a single major for the BCMS(Hons) or the BSc, students must gain 135 points from papers listed for Mathematics, including 105 points above 100 level, and 60 Mathematics at 500 level, including MATHS520.
To complete Mathematics as part of a double major for the BCMS(Hons), BSc or other undergraduate degree, students must gain 120 points from papers listed for Mathematics, including 90 points above 100 level, and 45 their first major at 500 level including MATHS520 if Mathematics is the first major.
To complete a minor in Mathematics, students must complete 60 points from the papers listed for the Mathematics major, including at least 30 points above 100 level.
100 levelBasic algebraic concepts and an introduction to Calculus and Statistics. This paper provides a last chance for students to correct a weak background in mathematics. Students who meet the prerequisites of MATH165 or MATH166 should take one of those papers instead.
This paper provides a first course in statistics for students in the Faculty of Science and Engineering. Microsoft Excel is used throughout. Topics include the collection and presentation of data, basic principles of experimental design, hypothesis testing, regression and the analysis of categorical data.
An introduction to statistical data collection and analysis. Topics include general principles for statistical problem solving; some practical examples of statistical inference; and the study of relationships between variables using regression analysis.
This paper introduces students to the R programming language which is used to investigate a collection of real data sets. Analysis of variance, multiple regression, non parametric methods and time series are covered.
This paper introduces statistical methods from a Bayesian perspective, which gives a coherent approach to the problem of revising beliefs given relevant data. It is particularly relevant for data analytics, statistics, mathematics and computer science students intending to enrol full-time in the first year of an undergraduate degree, majoring in Mathematics, in the Faculty of Computing & Mathematical Sciences. Students will sit a Scholarship exam. The Scholarship will have a value of up to $5,000.
For students who are enrolled or intending to enrol full-time in the research portion (i.e. a 90- or 120-point thesis) of their Masters degree in Mathematics in the year of application. It is expected that the bulk of the research will take place in the year of application. The Scholarship has a value of up to $5,000.
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Reading Comprehension
Math
Algebra
English Grammar
Geometry
How to Write an Essay
And a lot more!
Hundreds of pages of review and tutorials on all topics, with step-by-step example solutions.
Easy to read tutorials - Main points are highlighted, followed by a detailed explanation. You can read and absorb fast!
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Mathematics Department
토픽 개요
Mathematics Department
Mathematics Department
What is Mathematics and Statistics about?
Mathematics is the exploration and use of patterns and relationships in quantities, space, and time. Statistics is the exploration and use of patterns and relationships in data. These two disciplines are related but different ways of thinking and of solving problems. Both equip students with effective means for investigating, interpreting, explaining, and making sense of the world in which they live.
Mathematicians and statisticians use symbols, graphs, and diagrams to help them find and communicate patterns and relationships, and they create models to represent both real-life and hypothetical situations. These situations are drawn from a wide range of social, cultural, scientific, technological, health, environmental, and economic contexts.
Why study Mathematics and Statistics?
By studying mathematics and statistics, students develop the ability to think creatively, critically, strategically, and logically. They learn to structure and to organise, to carry out procedures flexibly and accurately, to process and communicate information, and to enjoy intellectual challenge.
By learning mathematics and statistics, students develop other important thinking skills. They learn to create models and predict outcomes, to conjecture, to justify and verify, and to seek patterns and generalisations. They learn to estimate with reasonableness, calculate with precision, and understand when results are precise and when they must be interpreted with uncertainty. Mathematics and statistics have a broad range of practical applications in everyday life, in other learning areas, and in workplaces.
Learning Area Structure:
The achievement objectives are presented in three strands. It is important that students can see and make sense of the many connections within and across these strands.
Number and algebra – Number involves calculating and estimating, using appropriate mental, written, or machine calculation methods in flexible ways. It also involves knowing when it is appropriate to use estimation and being able to discern whether results are reasonable. Algebra involves generalising and representing the patterns and relationships found in numbers, shapes, and measures.
Geometry and measurement– Geometry involves recognising and using the properties and symmetries of shapes and describing position and movement. Measurement involves quantifying the attributes of objects, using appropriate units and instruments. It also involves predicting and calculating rates of change.
Statistics involves identifying problems that can be explored by the use of appropriate data, designing investigations, collecting data, exploring and using patterns and relationships in data, solving problems, and communicating findings. Statistics also involves interpreting statistical information, evaluating data-based arguments, and dealing with uncertainty and variation.
PNBHS Mathematics Department
Level 1 Mathematics
This course covers material up to Level 6 of the National Curriculum, encompassing the three strands – Number & Algebra, Geometry & Measurement and statistics.
Upon completion, students will be fully prepared for the Level 1 NCEA examination.
A predominantly internally assessed alternative course is also available.
Mathematics (20 Credits)
Prerequisites: 14 Credits in Level Two Mathematics or a Level Two NCEA certificate.
This course has been designed for students who have not met the entry requirements for Calculus but would like to continue with Mathematics rather than a pure Statistics course at Level Three. This course covers the basics of Operational Research.
Stats Chat
Data visibility as a political act: Ben Goldacre and co-conspirators have set up a webpage tracking clinical trials that are violating the FDA Amendment Act (2007) by not having reported any results. It only became possible to violate the Act this Monday, so the compliance is fairly high so far, nearly 90%. Politician Sam is […]
Team Ratings for Round 2 The basic method is described on my Department home page. Here are the team ratings prior to this week's games, along with the ratings at the start of the season. Current Rating Rating at Season Start Difference Hurricanes 16.18 16.18 0.00 Crusaders 15.23 15.23 0.00 Lions 13.25 13.81 -0.60 […]
The theme of this year's Ihaka Lecture Series is "A thousand words: Visualising statistical data". The distillation of data into an honest and compelling graphic is essential component of modern (data) science, and this year, we have three experts exploring different facets of data visualisation. Each event begins at 6pm in the Large Chemistry Lecture Theatre, Building […]
There's a viral story that viral stories are shared by people who don't actually read them. I saw it again today in a tweet from Newseum Insititute 59% of links shared on social media are never actually clicked, according to a Columbia University study. Which means the majority of people don't actually read an article […]
There's a fun visualisation in the Herald of best places in NZ to retire. Chris Knox's design lets you adjust the relative importance of a set of factors, and also see which factors are responsible for a good or bad ranking for your favorite region. For nerds, he's even put up the code and data. […] | 677.169 | 1 |
Article/Info
An "Academic Term" or "Term" is a division of an academic year, the time during which an educational institution holds classes. The schedules adopted may vary widely. A "Trimester" (from the Latin meaning "three-monthly") divides the academic year into four terms, up to 12 weeks each and generally, the summer is counted as one of the terms. That simply means, we split our year into four terms. Students will get 3 Months to complete one term of any of the terms that are available.
When we think about Terms, we think what is good for the student, and what is good for the people who really want to learn Mathematics. Hence the decision to divide the books into 4 terms as being best for the students.
Do not be daunted by the 4 Term presentation of the workbooks as each of the Term Workbooks are easy to follow and easily understood.
Term 1 starts in January and ends in March.
Term 2 starts in April and ends in June.
Term 3 starts in July and ends September.
Term 4 starts in October and ends December.
Students can discuss any problems they might have in our forum. They can read the blog post. They will get mobile updates via our ROM apps as they become available. And there are also email update systems that can help the student to learn and increase their knowledge and understanding of math. | 677.169 | 1 |
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Book review
Written for a wide-range of mathematical abilities, this comprehensive, accessible overview emphasizes the conceptual aspects of decision-making rather than mathematical techniques or computer methods. Structured around a clear framework, the text first b | 677.169 | 1 |
LIVINGSTON
Hattie Criswell
February 19 – 23, 2018
Algebra – This week continue our study of quadratic equations focusing vertex form and standard form of quadratics equations and changing between the two forms. We will also be reviewing simplifying radicals, the square root method, and the quadratic formula.
8th Grade – We will review angle relationships and Pythagorean Theorem. We will have our Unit 9 Test on Friday 2/23. Our Mock Math STAAR will be next week. | 677.169 | 1 |
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