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Precalculus - 5th edition
Summary: Precalculus, Fifth Edition, by Lial, Hornsby, Schneider, and Daniels, engages and supports students in the learning process by developing both the conceptual understanding and the analytical skills necessary for success in mathematics. With the Fifth Edition, the authors adapt to the new ways in which students are learning, as well as the ever-changing classroom environmentPlease read description before purchase>> annotated teacher edition with publisher notation "review copy.." on cover New text no writing or marks includes all Students content and all answers. text on...show morely no access code or other supplements. ship immediately - Expedited shipping available ...show less
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Book summary
A prominent mathematician presents the principal ideas and methods of number theory within a historical and cultural framework. Oystein Ore's fascinating, accessible treatment requires only a basic knowledge of algebra. Topics include prime numbers, the Aliquot parts, linear indeterminate problems, congruences, Euler's theorem, classical construction problems, and many other subjects. |
Beginning Math Concepts for Game Developers
9781598632903
ISBN:
1598632906
Edition: 1 Pub Date: 2006 Publisher: Course Technology
Summary: Successful game programmers understand that in order to take their skills beyond the basics, they must have an understanding of central math topics; however, finding a guide that explains how these topics relate directly to games is not always easy. Beginning Math Concepts for Game Developers is the solution! It includes several hands-on activities in which basic math equations are used for the creation of graphs and..., ultimately, animations. By the time you finish, you will have developed a complete application from the ground up that allows you to endlessly explore game development scenarios for 2D games. If you have a basic understanding of programming essentials and a desire to hone your math skills, then get ready to take a unique journey that examines what is possible when you combine game development with basic math concepts.
Flynt, John P. is the author of Beginning Math Concepts for Game Developers, published 2006 under ISBN 9781598632903 and 1598632906. Two hundred sixty four Beginning Math Concepts for Game Developers textbooks are available for sale on ValoreBooks.com, one hundred eleven used from the cheapest price of $3.50, or buy new starting at $21.85 |
Algebra Course Assistant
Description This app covers the following topics applicable to Algebra I, Algebra II, and College Algebra:
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- Plot basic, parametric, or polar plots of the function(s) of your choice.
- Expand any polynomial.
- Factor numeric expressions, polynomials, and symbolic expressions.
- Divide any two expressions.
-Users Also Installed
Calculus Course Assistant
Wolfram Alpha, LLC
189 ratings
Taking calculus? Then you need the Wolfram Calculus Course Assistant. This definitive app for calculus--from the world leader in math software--will help you work through your homework problems, ace your tests, and learn calculus concepts. ForgetUmask Calculator
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Umask Calculator helps you calculate UNIX and Linux permissions when files and directories are created using a umask. * Completely free
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This course is a rigorous analysis of the real numbers, as well as an
introduction to
writing and communicating mathematics well.
Topics will include: construction of the real numbers, fields,
complex numbers, topology of the reals, metric spaces, careful
treatment of sequences and series, functions of real numbers, continuity,
compactness, connectedness, differentiation, and the mean value
theorem, with an introduction to sequences of functions.
It is the first course in the analysis sequence, which
continues in Real Analysis II.
Goals of the course:
Learn the content of real analysis.
Learn to read and write rigorous proofs.
Learn good mathematical writing skills and style.
Required Text:
Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill.
We will cover Chapters 1 through 5. There are also many other books
on analysis that you may wish to consult in the library, around the
QA300 area.
Homeworks, and Re-Writes:
Due at my office (Olin 1269) by 5:15pm on Thursdays.
Because I want you
to learn from the feedback you get on your homework, as well as
improve your writing skills, I will use a system of (optional)
re-writes for the first few assignments, which will work as follows:
Turn in the homework by 5:15pm on the due date.
The homework will be graded and returned to you within one week.
If you are not satisfied with the grade you received on the
homework, you have the option of re-doing any question(s) you
wish, and submitting the re-written version together with the
previously graded version. (You may only re-write a question if
you made a serious attempt at it on the first version.)
If you choose to do a re-write, it is due at my office by 5:15pm
two weeks after the original due date of the assignment.
Your re-write will be graded, with particular attention to whether
you adopted the graders' suggestions, and new grades will be assigned
for rewritten questions. Your grade for a rewritten question will
always go up or stay the same; it will never go down.
Rewrites will only be accepted for Homeworks 1 through 4.
See also this guide.
Late homeworks can only be accepted by special permission.
Please ask at least 24 hours in advance. The lowest
homework assignment will be dropped. Please follow the HMC
Mathematics Department format for homework, online at
LaTeX: some of you may find LaTeX helpful in typesetting your
homework. If you'd like to learn LaTeX, or have questions about it,
you can visit the CCMS
Software Lab.
Midterms and Grading:
There will be three mid-terms:
Handed out: Mon Feb 25, due Thu Feb 28.
Handed out: Mon Apr 1, due Thu Apr 4.
Thu May 9 for seniors and Monday 9am-noon May 13 for everyone else.
Each of these and your homework average will count 25% of your course
grade.
Honor Code:
The HMC Honor Code applies in all matters of conduct concerning this
course. Though cooperation on homework assignments is encouraged, you
are expected to write up all your solutions individually. Thus
copying is prohibited, and
you should understand your solutions well enough to write them up
yourself. It is appropriate to acknowledge the assistance of others;
if you work with others on a homework question, please write their
names in the margin.
Note that some of the homework problems in this course have been
assigned in prevous semesters. Copying work from published solutions
(or solutions of past students) is a violation of the HMC Honor Code
and will be handled accordingly.
Taped YouTube Lectures:
These lectures were taped in 2010, and although the lectures I give
this year may not be identical, they will be close enough that you may
find it valuable to use them for review. Or, better yet, watch them
before the class lecture, and then during class you can ask questions!
(The taped classes were 75 minutes long, so the lecture numbers will
not line up with this year's MWF 50 minute classes.)
1. What is a good rule of thumb for what you should assume of
your audience as you write your homework sets?
2. Is blackboard writing formal or informal writing?
3. Do you see why the proof by contradiction on page 3 is not really a
proof by contradiction?
4. Name 3 things a lazy writer would do that a good writer wouldn't.
5. What's the difference in meaning between these three phrases?
"Let A=12."
"So A=12."
"A=12."
Now examine Section 1.1 of Rudin, showing that there is no rational p
that
satisfied p2=2.
6. There are many places in his proof where he could have used symbols
to
express his ideas, but he does not. (e.g., "Let A be the set of
all
positive rationals p such that...") Why do you think he chooses
not to
use symbols?
7. What would you change about his presentation if you were writing
for a
high school audience? Give a specific example.
HW #1. Due Jan 31. Read Chapter 1 in its entirety. Do not worry
about understanding everything, just read for the big ideas. Turn in
the following problems:
Problem A. In no more than four sentences,
describe the main themes and concepts of Chapter 1. The first
sentence or two should be at a level that your parent could
understand even if they never went to college.
The other sentences should be understandable by any college student.
Problem B. Recall that in class, we defined a rational
number m/n to be an equivalence class of pairs (m,n) under an
equivalence relation. Check that this equivalence relation is
transitive: if (p,q)~(m,n) and (m,n)~(a,b), then (p,q)~(a,b).
Problem C.
We defined addition of rational numbers in
terms of representatives: a/b + c/d = [ad+bc]/[bd].
Show that the addition of rational numbers is well-defined.
Problem D. Define a multiplication of rational numbers, and show this
multiplication is well-defined.
p.s. Be sure to subscribe to math-131-l (using listkeeper@hmc.edu) if
you are not already getting e-mails from the list.
HW #2. Due Feb 7.
A problem marked "R" means read, but do not do the problem.
Chapter 1 ( R3bcd, 4, 5, R7, 8, 9, 20 ) and
Problem S. For a real number a
and non-empty
subset of reals B, define: a + B =
{ a + b : b is in B }. Show that if B is bounded above, then sup( a + B ) = a + sup B.
[For 1.20, whenever the proof is
EXACTLY THE SAME as in Steps 3 and 4 of pp. 18-19, you do not need
to
re-write the proof. Just point out that the proof is the same as
in
Rudin's. But wherever the proof differs, BE SURE to POINT OUT HOW
IT DIFFERS, and VERIFY all new things.]
Rewrites for HW#1 are also due.
If you want to do rewrites,
see this guide.
HW #4. Due Feb 21.
A problem marked "R" means read, but do not
do the problem.
Do Chapter 2 ( 2, R3, 4, 5, R6, R7, 8, 11[exclude d2] ).
Problem T. Prove that the Principle of Induction implies the
Well-Ordering Principle for N (the natural numbers).
[Hint: Let
L(n)
be the statement "if A is a subset of N that contains
a number <= n, then A has a least element".
Now prove L(n) holds for all n by induction.]
Rewrites for HW#2 are also due.
If you want to do rewrites,
see this
guide. Save a copy since you probably won't get the homework
back in time to study for the exam.
EXAM 1. Available Monday Feb 25 (afternoon).
Though test is due FRIDAY Mar 1 at 2:30pm.
Rewrites for HW#3 are still due Thursday 2/28.
HW #5. Due Mar 7. A problem marked "R" means read, but do not
do the problem. There are no re-writes on this homework and forward.
Do Chapter 2 ( 9ab, 9cd, 9ef, R10, 12, R14, 16, 22, R23 ).
In Problem 14, give an example of a cover that is not a
nested collection of open sets.
Rewrites for HW#4 are also due.
HW #6. Due Mar 14. A problem marked "R" means read, but do not
do the problem. (I've split up some problems into pieces to indicate
which pieces are worth a similar amount of points, and the graders
will grade these parts separately.)
HW #10. Due Apr 25.
A problem marked "R" means read, but do not do the problem.
Do Chapter 4 ( 9*, 10, 11**, 12, 14, R16, 18, R19 ).
*The definition of diameter can be found on page 52.
**You do NOT have do the 2nd part of Problem 4.11, where it says "Use
this result to give..."
Please come see me or the tutors to discuss these problems!
HW #11. Due May 2.
Do Problem E. Use the mean value theorem to show that e^x is greater
than or equal to (1+x) for all x in R. (You may assume knowledge of
the derivative of e^x.)
and Chapter 5 ( 1, 4, 13ab*, R13cdefg, R25abd, 25c )
and Chapter 7 ( 1, R2, R3 ).
*In Problem 5.13, there is a typo (in some editions of the book): the
xa should say |x|a. |
Web Codes
Digital Solutions
Prentice Hall High School Mathematics Digital Classroom Resources
Technology tools to Enhance, Engage, and Enrich the teaching and learning experience.
Check out the technology ranging from online activities, videos, slideshows and much more.
Resources include:
Success Tracker Interactive Textbook
Prentice Hall's interactive textbooks offer the same trusted content as your print textbooks, making it easy to transition from textbook to technology. The interactive textbook, available online and on CD-ROM, engages and motivates all learners with:
Success Tracker™ is an online, formative assessment and remediation management system. It makes data-driven instruction with personalized remediation practical for all your students through time-saving, automated tools. |
Mathematical Physics110.41
FREE
About the Book
For physics students interested in the mathematics they use, and for math students interested in seeing how some of the ideas of their discipline find realization in an applied setting. The presentation strikes a balance between formalism and application, between abstract and concrete. The interconnections among the various topics are clarified both by the use of vector spaces as a central unifying theme, recurring throughout the book, and by putting ideas into their historical context. Enough of the essential formalism is included to make the presentation self-contained. |
elementary text introduces basic quantum mechanics to undergraduates with no background in mathematics beyond algebra. Containing more than 100 problems, it provides an easy way to learn part of the quantum language and apply it to problems. Emphasizing the matrices representing physical quantities, it describes states simply by mean values of physical quantities or by probabilities for possible values. This approach requires using the algebra of matrices and complex numbers together with probabilities and mean values, a technique introduced at the outset and used repeatedly. Students discover the essential simplicity of quantum mechanics by focusing on basics and working only with key elements of the mathematical structure--an original point of view that offers a refreshing alternative for students new to quantum mechanics.
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--This text refers to an out of print or unavailable edition of this title.
Most Helpful Customer Reviews
This review is written from the point of view of a philospher, poorly trained in mathematics, but still wanting to get to the meat of quantum mechanics from a methematical point of view. Wow. In this book I found what I thought I never would. It describes the mathematical world of quantum physics using the majestic simplicity of matrices and the algebra of complex numbers. As the author states in the preface, no calculus or trigonometry is required. While the math isn't downright simple, neither is beyond the grasp of someone who is bright, but hasn't taken claculus or even precalc. For those who want to journey past this book another excellent intro level quantum mechanics text that introduces wave mechanics and does assume a knowledge of basic calculus is "Fundamentals of Quantum Mechanics" by J.E. House. Both are excellent!
There are few books which explain quantum mechanics with such grace and simplicity. Starting with the basics the author sets out to explain the ideas and mathematics behind qunatum mechanics. The author also provides the historical references leading to the birth of quantum mechanics. The layout and presentation of the material is pure mathematical poetry. Whilst the material would never make light bedtime reading, I would seriously recommend this book for both phyisicists and electronic engineering at the undergraduate and graduate level. The book has been a great source of information for my own research into the mysteries of quantum mechanics.
I have quite a few books on Quantum Mechanics. This book does what the others do not. The first half is about simple math. Understanding that QP - PQ = ih/2pi is the matrix form of an equation and the QP - PQ is not zero because the matrices do not commute is critical. This is basic stuff that a lot of books just skip. The second half uses the math to explain some of the features of Quantum Mechanics. For me I needed the detailed first half even though the math was not too hard. Now I can read my other books with a new understanding and finally I am starting to understand Quantum Mechanics.
I liked this book and learned a good deal from it. It is intended as a look at only some aspects of QM -- a slice -- not the subject as a whole. It has some problems: he never defines quite why or how the given matrices are chosen for ecample. It seems like a good "add on" to whatever other introduction to QM you are reading.
The one star is not for the text, but for the quality of the paper and printing. Dover used to dependably print their technical books on good quality paper, but this book had astonishingly bad paper, and the ink bleeds into it. At least, I won't feel bad marking it up... They seem to have used the paper they use on their one dollar classic novel reprints, or worse.
This book is intended to introduce quantum mechanics to beginners at the level of a Scientific American article. No knowledge of calculus is assumed; the reader can probably get by with nothing more than high school Algebra II. Due to these constraints, a great deal of material must necessarily be left out. As calculus is not used in this book, there is no mention of the Schrodinger equation or differential operators.
When I was a college physics major first learning modern physics 40 years ago, it was not until I encountered the solution to the hydrogen atom using the Schrodinger equation that I began to feel comfortable with quantum mechanics. Schrodinger's solution to the hydrogen atom was sufficiently specific and detailed to show the power of quantum theory, and in historical terms provided the theory with much needed credibility. I would therefore suggest that the reader with knowledge of calculus read an introductory book that includes the Schrodinger equation, such as Cropper's The Quantum Physicists: And an Introduction to Their Physics.
Very good ! The book helps the reader very well to understand the way of thinking that he needs to get to use when dealing with quantum mechanics. The book gives the opportunity to get the explanations directly from the source - the people who were facing the question why a new mechanic is necessary and what are the things that make it a new mechanic compared to the classical mechanic.
It's true that this book requires absolutely no calculus. Or linear algebra for that matter. This book doesn't even assume you've ever seen a complex number or a matrix before. All that is necessary is introduced in the first few chapters.
However, as this book progresses it slowly reveals itself for what it truly is: a first book on the operator formalism in quantum mechanics, where commutation relations for observable quantities are promoted to central importance.
While I'm certain that students with only a very modest background in physics and mathematics will be able to get something out of this book at least in the early chapters, the last third of this book is more suitable for fairly advanced students of quantum mechanics looking to make their way from state vectors to operators as required by quantum field theory. To such students I would recommend already having The Principles of Quantum Mechanics (International Series of Monographs on Physics) under your belt.
This is ultimately a challenging book masquerading as an elementary one. |
Transition from arithmetic to algebra. Includes signed numbers, commutative, associative, and distributive laws, order of operations, algebraic expressions, polynomials, fractions, and linear equations. Also includes percents, ratio and proportion, graphing, perimeter, area, volume, and optional topics.
Prerequisite(s): Within the last three years: MAT 082 with a C or better or required score on the Mathematics assessment test.
Information: The course content is offered in 35 modules which are computer delivered in a structured, individualized learning environment with on-demand instruction assistance. Attendance at regularly scheduled classes is required. The course may be taken four times for a maximum of twelve credit hours. Most students will take this course more than once. To earn a passing grade, students must successfully complete a minimum of 9 modules. Upon successful completion of module 35, a student's enrollment is converted to and credit is received for Mat 122Z.
Information: Upon completion of all modules of MAT 089, students will have met all of the competencies of MAT122 and will receive credit equivalent to MAT 122Z. No more than 3 credit hours can be applied toward graduation for MAT 122, 122Z, and/or 123.
Prerequisite(s): Within the last three years: C or better in MAT 092 or satisfactory score on the mathematics assessment exam.
Course Corequisites:
Information: No more than 3 credit hours can be applied toward graduation for MAT 122, 122Z and/or 123. Information: Access to a scanner required for Math classes taken online. Information: Not a university level course.
College Algebra with an emphasis on data analysis. Includes functions, systems of equations, exponents and logarithms, power functions, polynomial functions, rates of change, descriptive statistics, regression, summation notation, spreadsheet software, and reports and projects.
Prerequisite(s): Within the last three years: MAT 122, 122Z or 123 with a C or better, or required score on the mathematics assessment test.
Course Corequisites:
Information: Basic computer skills are required and may be attained through CSA 101 or 110 or 110A. See an advisor or mathematics faculty member for information.
An overview of mathematical concepts, principles and applications specifically for elementary teachers. Includes real number properties and patterns, arithmetic operations and algorithms in subsets of real numbers, alternative numbers systems, set theory, and algebraic reasoning and problem solving. Also includes the technology to teach mathematics.
Prerequisite(s): Within the last three years: MAT 142 or 144 or 151 or higher with a C or better, or mathematics assessment into MAT 167 or higher.
Course Corequisites:
Information: Access to a scanner required for math classes taken online.
Prerequisite(s): Within the last three years: MAT 122, 122Z, or 123 with a C or better, A graphing calculator is required. See your instructor for details. Information: Access to a scanner required for Math classes taken online.
Introduction to statistics. Includes the nature of statistics, quantitative data, probability, probability distributions and the central limit theorem. Also includes estimates for population parameters, hypothesis testing, correlation with regression, and additional topics with choices from chi square distribution, ANOVA and/or nonparametric methods.
Prerequisite(s): Within the last three years: MAT 144 or 151 with a C or better, or required score on the Mathematics assessment test.
Course Corequisites:
Information: Use of a graphing calculator and/or computer programs may be required at the discretion of the instructor. Access to a scanner required for math classes taken online.
Prerequisite(s): Within the last three years: MAT 151 with a C or better Access to a scanner required is for math classes taken online.
Prerequisite(s): Within the last three years: MAT 122, 122Z, or 123 with a B or better, or required score on the Mathematics assessment test.
Course Corequisites:
Recommendation: Fast-paced course for motivated students who have strong algebra skills and some trigonometry background.
Information: Students taking two or three of the following courses in any combination will receive a maximum of 7 credits toward graduation: MAT 151, 182 and 187. Information: A graphing calculator will be required for this course and will be used extensively.
Introduction to differential equations. Includes first order differential equations, higher order differential equations, systems of linear differential equations, Laplace transforms, and approximating methods. Also includes applications.
Prerequisite(s): Within the last three years: MAT 231 with a C or better. |
This course is intended for students who have a thorough knowledge of analytic geometry and elementary functions in addition to college preparatory algebra, geometry, and trigonometry. The purpose of the course is to prepare the student for advanced placement in college calculus. In this course:
In this course the student will
·Define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits.
·State the definition of continuity and determine where a function is continuous or discontinuous. This will include continuity at a point; continuity over a closed interval; application of the Intermediate Value Theorem; and graphical interpretation of continuity and discontinuity.
·Find the derivative of an algebraic function by using the definition of a derivative. This will include investigating and describing the relationship between differentiability and continuity.
·Apply formulas to find the derivative of algebraic, trigonometric, exponential, and logarithmic functions and their inverses.
·Apply formulas to find the derivative of the sum, product, quotient, inverse, and composite (chain rule) of elementary functions.
·Find the derivative of an implicitly defined function.
·Find the higher order derivatives of algebraic, trigonometric, exponential, and logarithmic functions.
·State (without proof) the Mean Value Theorem for derivatives and apply it both algebraically and graphically.
·Use L'Hopital's rule to find the limit of functions whose limits yield the indeterminate forms: 0/0 and infinity/infinity. A Calculus, these functions will also include functions whose limits yield the indeterminate forms: 0 to the 0th power, 1 to the infinity power, infinity to the infinity power, infinity minus infinity.
·Apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, and optimization problems. |
BasicElectricalFormulas. INTRODUCTION TO UNIT 1—ELECTRICIAN'S MATH AND BASICELECTRICALFORMULAS. In order to construct a building that will last into the future, a strong foundation is a prerequisite. The foundation is a part of the building that
"ElectricalEngineering Overview" Prepared as part of the Sloan Career Cornerstone Center ... The basic functions of engineering are defined by the sequencing of engineering work: research, design and development, testing, manufacturing, construction, service and maintenance, and management.
... basicelectrical skills needed for him/her to safely and more efficiently carry out their duties in the plant environment. After successful ... SOLVING EQUATIONS AND FORMULAS ... POWERS OF TEN NOTATION USING ENGINEERING UNITS Powers of ten notation follows the same rules for ...
The basic mathematics knowledge the graduates have acquired in the College is valuable for solving related problems encountered in their various technical ... 1.4 Simultaneous differential equations and their applications in electricalengineering. 2.
provided mainly in support of the EC module D227 – Control System Engineering. This tutorial is mainly descriptive. ... In order to complete the theoretical part of this tutorial, you must be familiar with basic mechanical and electrical science.
modern electricalengineering, a realization made over a century ago. 2.1.1 De nitions The notion of the square root of −1 originated with the quadratic formula: the solution of certain quadratic ... Equivalent circuits can be used in two basic ways.
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The rest of the formulas are basicelectricalengineeringformulas. In the formula evaluating process online calculation tool Wolphram Alpha was used. As a circuit simulator 5SPICE was used. Due to the algorithm complexity there is a necessity for the iterative process.
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The study of electricalengineering theory is extremely complex. Fortunately, the practical appli- ... ElectricalFormulas The following basicformulas govern the relation-ship of voltage, amperage, and power in electrical circuits. 2
Electrical Circuits A complete path, or circuit, is needed before voltage can cause a current flow through resistances to perform work. ... Here are some basicformulas you will find helpful in solving more complex electrical problems. They |
Computer Algebra and Symbolic Computation: Elementary Algorithms provides a systematic approach for the algorithmic formulation and implementation of...... more
Computer Algebra and Symbolic Computation: Elementary Algorithms provides a systematic approach for the algorithmic formulation and implementation of mathematical operations in computer algebra programming languages. The viewpoint is that mathematical expressions, represented by expression trees, are the data objects of computer algebra programs, and by using a few primitive operations that analyze and construct expressions, we can implement many elementary operations from algebra, trigonometry, calculus, and differential equations. With a minimum of prerequisites, this book is accessible and useful to students of mathematics, computer science, and other technical fields. The book contains a CD with the full, searchable text and implementations of all algorithms in Maple, Mathematica, and MuPad programming languages.
Analytical Methods in Anisotropic Elasticity Focuses on the mathematical techniques and solution methodologies required to establish the foundations of...... more
Analytical Methods in Anisotropic Elasticity Focuses on the mathematical techniques and solution methodologies required to establish the foundations of anisotropic elasticity and provides the theoretical background for composite material analysis. This work is useful for graduate students and professors, applied mathematicians, numerical analysts, scientists, engineers, and researchers. Full description
These black and white large picture cards can be used in so many ways - add a special touch to party decorations and presents - use them as mini cards or notes...... more
These black and white large picture cards can be used in so many ways - add a special touch to party decorations and presents - use them as mini cards or notes - as an educational learning resource for activities such as patterning and sequencing - place on calendars to mark special days and events - and so much more!
Reviews »
...If you loved the Da Vinci Code, you'll enjoy this. Conversely, if you've found Dan Brown's work to be piles of bombastic tosh, you'll most likely feel much the same way about the Lost Symbol.
Dan Brown is many things as an author - insert your choice of adjectives as appropriate. One quality he does possess amongst this array is reliability; if you appreciate fast-paced narratives that read...
...'The Lost Symbol is the latest offering from Dan Brown featuring Harvard Symbologist Robert Langdon and like all avid readers of his work I was looking forward to reading the book.
-
At 6am on a Sunday morning Robert Langdon receives a message from his mentor and friend, prominent 33rd Degree Mason Peter Solomon, to conduct a lecture in Washington?s Capitol Building and not wanting to let...
...Book Info:
Name: The Lost Symbol
Author: Dan Brown
Released: September 2009
Series: Robert Langdon Stories
Awards: N/A
I think one of the best qualities I love so much in books is their unique ability to diversify into so many genres, so many avenues to flow through, one being a hilarious family adventure, another being a steamy, albeit cheesy... |
Thinking Mathematically, Books a la Carte Edition, 5th Edition
Description
This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Books à la Carte also offer a great value—this format costs significantly less than a new textbook.
Blitzer continues to raise the bar with his engaging applications developed to motivate readers from diverse majors and backgrounds. Thinking Mathematically, Fifth Edition, draws from the author's
Table of Contents
1. Problem Solving and Critical Thinking
1.1 Inductive and Deductive Reasoning
1.2 Estimation, Graphs, and Mathematical Models
1.3 Problem Solving
2. Set Theory
2.1 Basic Set Concepts
2.2 Subsets
2.3 Venn Diagrams and Set Operations
2.4 Set Operations and Venn Diagrams with Three Sets
2.5 Survey Problems
3. Logic
3.1 Statements, Negations, and Quantified Statements
3.2 Compound Statements and Connectives
3.3 Truth Tables for Negations, Conjunction, and Disjunction
3.4 Truth Tables for the Conditional and the Biconditional
3.5 Equivalent Statements and Variations of Conditional Statements
3.6 Negations of Conditional Statements and De Morgan's Laws
3.7 Arguments and Truth Tables
3.8 Arguments and Euler Diagrams
4. Number Representation and Calculation
4.1 Our Hindu-Arabic System and Early Positional Systems
4.2 Number Bases in Positional Systems
4.3 Computation in Positional Systems
4.4 Looking Back at Early Numeration Systems
5. Number Theory and the Real Number System
5.1 Number Theory, Prime and Composite Numbers
5.2 The Integers; Order of Operations
5.3 The Rational Numbers
5.4 The Irrational Numbers
5.5 Real Numbers and Their Properties
5.6 Exponents and Scientific Notation
5.7 Arithmetic and Geometric Sequences
6. Algebra: Equations and Inequalities
6.1 Algebraic Expressions and Formulas
6.2 Linear Equations in One Variable and Proportions
6.3 Applications of Linear Equations
6.4 Linear Inequalities in One Variable
6.5 Quadratic Equations
7. Algebra: Graphs, Functions, and Linear Systems
7.1 Graphing and Functions
7.2 Linear Functions and Their Graphs
7.3 Systems of Linear Equations in Two Variables
7.4 Linear Inequalities in Two Variables
7.5 Linear Programming
7.6 Modeling Data: Exponential, Logarithmic, and Quadratic Functions
8. Consumer Mathematics and Financial Management
8.1 Percent, Sales Tax, and Income Tax
8.2 Simple Interest
8.3 Compound Interest
8.4 Annuities, Stocks, and Bonds
8.5 Installment Loans, Amortization, and Credit Cards
9. Measurement
9.1 Measuring Length; The Metric System
9.2 Measuring Area and Volume
9.3 Measuring Weight and Temperature
10. Geometry
10.1 Points, Lines, Planes, and Angles
10.2 Triangles
10.3 Polygons, Perimeter, and Tessellations
10.4 Area and Circumference
10.5 Volume
10.6 Right Triangle Trigonometry
10.7 Beyond Euclidean Geometry
11. Counting Methods and Probability Theory
11.1 The Fundamental Counting Principle
11.2 Permutations
11.3 Combinations
11.4 Fundamentals of Probability
11.5 Probability with the Fundamental Counting principle, Permutations, and Combinations
11.6 Events Involving Not and Or; Odds
11.7 Events Involving And; Conditional Probability
11.8 Expected Value
12. Statistics
12.1 Sampling, Frequency Distributions, and Graphs
12.2 Measures of Central Tendency
12.3 Measures of Dispersion
12.4 The Normal Distribution
12.5 Problem Solving with the Normal Distribution
12.6 Scatter Plots, Correlation, and Regression Lines
13. Mathematical Systems
13.1 Mathematical Systems
13.2 Rotational Symmetry, Groups, and Clock Arithmetic
14. Voting and Apportionment
14.1 Voting Methods
14.2 Flaws of Voting Methods
14.3 Apportionment Methods
14.4 Flaws of Apportionment Methods
15. Graph Theory
15.5 Graphs, Paths, and Circuits
15.2 Euler Paths and Euler Circuits
15.3 Hamilton Paths and Hamilton Circuits
15.4 Trees
This title is also sold in the various packages listed below. Before purchasing one of these packages, speak with your professor about which one will help you be successful in your course. |
More About
This Textbook
Overview
Elementary Linear Algebra 10th edition gives an elementary treatment of linear algebra that is suitable for a first course for undergraduate students. The aim is to present the fundamentals of linear algebra in the clearest possible way; pedagogy is the main consideration. Calculus is not a prerequisite, but there are clearly labeled exercises and examples (which can be omitted without loss of continuity) for students who have studied calculus. Technology also is not required, but for those who would like to use MATLAB, Maple, or Mathematica, or calculators with linear algebra capabilities, exercises are included at the ends of chapters that allow for further exploration using those tools.
A concluding chapter covers twenty applications of linear algebra drawn from business, economics, physics, computer science, ecology, genetics, and other disciplines. The applications are independent and each includes a list of mathematical prerequisites |
Differential Equations
9780495012658
ISBN:
0495012653
Edition: 3 Pub Date: 2005 Publisher: Thomson Learning
Summary: Incorporating a modeling approach throughout, this exciting text emphasizes concepts and shows that the study of differential equations is a beautiful application of the ideas and techniques of calculus to everyday life. By taking advantage of readily available technology, the authors eliminate most of the specialized techniques for deriving formulas for solutions found in traditional texts and replace them with topi...cs that focus on the formulation of differential equations and the interpretations of their solutions. Students will generally attack a given equation from three different points of view to obtain an understanding of the solutions: qualitative, numeric, and analytic. Since many of the most important differential equations are nonlinear, students learn that numerical and qualitative techniques are more effective than analytic techniques in this setting. Overall, students discover how to identify and work effectively with the mathematics in everyday life, and they learn how to express the fundamental principles that govern many phenomena in the language of differential equations.
Devaney, Robert L. is the author of Differential Equations, published 2005 under ISBN 9780495012658 and 0495012653. One hundred thirty eight Differential Equations textbooks are available for sale on ValoreBooks.com, twenty four used from the cheapest price of $20.48, or buy new starting at $186.51 |
...
Discrete Mathematics and its Applications, Sixth mathematics to a wide a wide variety of real-world applications…from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields.
Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, FOURTH
While learning about such conceptsThis book is a clear and self-contained introduction to discrete mathematics. Aimed mainly at undergraduate and early graduate students of mathematics and computer science. It is written with the goal of stimulating interest in mathematics and an active, problem-solving approach to the presented material. The reader is led to an understanding of the basic principles and methods of actually doing mathematics (and having fun at that). Being more narrowly focused than many discrete mathematics textbooks and treating selected topics in an unusual depth and from several points of view, the book reflects the conviction of the authors, active and internationally renowned mathematicians, that the most important gain from studying mathematics is the cultivation of clear and logical thinking and habits useful for attacking new problems.
For one or two term introductory courses in discrete mathematics. This book provides an accessible introduction to discrete mathematics through an algorithmic approach that focuses on problem-solving techniques with woven techniques of proofs into the text as a running theme. Each chapter has a problem-solving corner that shows students how to attack and solve problems. The topics are based upon the recommendations of various professional organizations, including those of the MAAs Panel on Discrete Mathematics in the First Two Years, the NCTMs Principles and Standards for School Mathematics, and the CBMSs recommendations for the mathematical education of teachers.
Master discrete mathematics and ace your exams with this easy-to-use guide that reinforces problem-solving skills and reduces your study time! Students of discrete mathematics love Schaum's--the first edition of this book was a major bestseller--and this edition will show you why! Schaum's Outline of Discrete Mathematics lets you focus on the problems that are at the heart of the subject. It cuts your study time by eliminating the extraneous material that clutters up so many textbooks. ...
In recent years, airline practitioners and academics have started to explore new ways to model airline passenger demand using discrete choice methods. This book provides an introduction to discrete choice models and uses extensive examples to illustrate how these models have been used in the airline industry. These examples span network planning, revenue management, and pricing applications. Numerous examples of fundamental logit modeling concepts are covered in the text, including probability calculations, value of time calculations, elasticity calculations, nested and non-nested likelihood ratio tests, etc. The core chapters of the book are written at a level appropriate for airline practitioners and graduate students with operations research or travel demand modeling backgrounds. Given the majority of discrete choice modeling advancements in transportation evolved from urban travel demand studies, the introduction first orients readers from different backgrounds by highlighting major distinctions between aviation and urban travel demand studies. This is followed by an in-depth treatment of two of the most common discrete choice models, namely the multinomial and nested logit models. More advanced discrete choice models are covered, including mixed logit models and generalized extreme value models that belong to the generalized nested logit class and/or the network generalized extreme value class. An emphasis is placed on highlighting open research questions associated with these models that will be of particular interest to operations research students. Practical modeling issues related to data and estimation software are also addressed, and an extensive modeling exercise focused on the interpretation and application of statistical tests used to guide the selection of a preferred model specification is included; the modeling exercise uses itinerary choice data from a major airline. The text concludes with a discussion of on-going customer modeling research in aviation. "Discrete Choice Modelling and Air Travel Demand" is enriched by a comprehensive set of technical appendices that will be of particular interest to advanced students of discrete choice modeling theory. The appendices also include detailed proofs of the multinomial and nested logit models and derivations of measures used to represent competition among alternatives, namely correlation, direct-elasticities, and cross-elasticities. ...
Interval and Discrete Transforms with Application and Error Analysis (Pure and Applied Mathematics)
CRC Press 1992 | 825 | ISBN: 0824782526 | PDF | 17 Mb
This reference/text desribes the basic elements of the integral, finite, and discrete transforms - emphasizing their use for solving boundary and initial value problems as well as facilitating the representations of signals and systems.;Proceeding to the final solution in the same setting of Fourier analysis without interruption, Integral and Discrete Transforms with Applications and Error Analysis: presents the background of the FFT and explains how to choose the appropriate transform for solving a boundary value problem; discusses modelling of the basic partial differential equations, as well as the solutions in terms of the main special functions; considers the Laplace, Fourier, and Hankel transforms and their variations, offering a more logical continuation of the operational method; covers integral, discrete, and finite transforms and trigonometric Fourier and general orthogonal series expansion, providing an application to signal analysis and boundary-value problems; and examines the practical approximation of computing the resulting Fourier series or integral representation of the final solution and treats the errors incurred.;Containing many detailed examples and numerous end-of-chapter exercises of varying difficulty for each section with answers, Integral and Discrete Transforms with Applications and Error Analysis is a thorough reference for analysts; industrial and applied mathematicians; electrical, electronics, and other engineers; and physicists and an informative text for upper-level undergraduate and graduate students in these disciplines. ... |
Linear Algebra
9780201119497
ISBN:
0201119498
Publisher: Addison-Wesley Longman, Incorporated
Summary: Fraleigh and Beauregard's text is known for its clear presentation and writing style, mathematical appropriateness, and overall usability. Its inclusion of calculus-related examples, true/false problems, section summaries, integrated applications, and coverage of Cn make it a superb text for the sophomore or junior-level linear algebra course. This Third Edition retains the features that have made it successful over ...the years, while addressing recent developments of how linear algebra is taught and learned. Key concepts are presented early on, with an emphasis on geometry. KEY TOPICS: Vectors, Matrices, and Linear Systems; Dimension, Rank, and Linear Transformations; Vector Spaces; Determinants; Eigenvalues and Eigenvectors; Orthogonality; Change of Basis; Eigenvalues: Further Applications and Computations; Complex Scalars; Solving Large Linear Systems MARKET: For all readers interested in linear algebra.
Fraleigh, John B. is the author of Linear Algebra, published under ISBN 9780201119497 and 0201119498. Twenty four Linear Algebra textbooks are available for sale on ValoreBooks.com, twenty three used from the cheapest price of $0.01, or buy new starting at $3,272.97 |
1. What is GeoGebra?
GeoGebra is dynamic mathematics software that joins geometry, algebra and calculus. It is
developed for learning and teaching mathematics in schools by Markus Hohenwarter and an
international team of programmers.
1.1. Multiple Views for Mathematical Objects
GeoGebra provides three different views of mathematical objects: a Graphics View, a,
numeric Algebra View, and a Spreadsheet View. They allow you to display mathematical
objects in three different representations: graphically (e. g., points, function graphs),
algebraically (e. g., coordinates of points, equations), and in spreadsheet cells. Thereby, all
representations of the same object are linked dynamically and adapt automatically to
changes made to any of the representations, no matter how they were initially created.
Toolbar Toolbar
Help
Algebra View Spreadsheet
View
Graphics View
Input Bar
1.1.1. Graphics View
Using the construction tools available in the Toolbar you can do geometric constructions in
the Graphics View with the mouse. Select any construction tool from the Toolbar and read
the Toolbar Help (next to the Toolbar) in order to find out how to use the selected tool. Any
object you create in the Graphics View also has an algebraic representation in the Algebra
View.
6
Note: After activating the tool Move you are able to move objects in the Graphics View by
dragging them with the mouse. At the same time, their algebraic representations are
dynamically updated in the Algebra View.
Every icon in the Toolbar represents a toolbox that contains a selection of similar
construction tools. In order to open a toolbox, you need to click on the small arrow in the
lower right corner of the Toolbar icon.
Hint: Construction tools are organized by the nature of resulting objects or the functionality
of the tools. You will find tools that create different types of points in the Point Toolbox
(default icon ) and tools that allow you to apply geometric transformations in the
Transformation Toolbox (default icon ).
1.1.2. Algebra View
Using the Input Bar you can directly enter algebraic expressions in GeoGebra. After hitting
the Enter‐key your algebraic input appears in the Algebra View while its graphical
representation is automatically displayed in the Graphics View.
Example: The input f(x) = x^2 gives you the function f in the Algebra View and its
function graph in the Graphics View.
In the Algebra View, mathematical objects are organized as free and dependent objects. If
you create a new object without using any other existing objects, it is classified as a free
object. If your newly created object was created by using other existing objects, it is
classified as a dependent object.
Hint: If you want to hide the algebraic representation of an object in the Algebra View, you
may specify the object as an auxiliary object: Right click (Mac OS: Ctrl‐click) on the
corresponding object in the Algebra View and select Properties from the appearing Context
Menu. On tab Basic of the Properties Dialog you may specify the object as an Auxiliary
Object. By default, auxiliary objects are not shown in the Algebra View, but you can change
this setting by selecting the item Auxiliary Objects from the View menu.
Note that you are able to modify objects in the Algebra View as well: Make sure that you
activate the Move tool before you double click on a free object in the Algebra View. In
the appearing text box you can directly edit the algebraic representation of the object. After
hitting the Enter‐key, the graphical representation of the object will automatically adapt to
your changes.
If you double click on a dependent object in the Algebra View, a dialog window appears
allowing you to Redefine the object.
GeoGebra also offers a wide range of commands that can be entered into the Input Bar. You
can open the list of commands in the right corner of the Input Bar by clicking on the button
Command. After selecting a command from this list (or typing its name directly into the
Input Bar) you can press the F1‐key to get information about the syntax and arguments
required to apply the corresponding command.
7
1.1.3. Spreadsheet View
In GeoGebra's Spreadsheet View every cell has a specific name that allows you to directly
address each cell. For example, the cell in column A and row 1 is named A1.
Note: These cell names can be used in expressions and commands in order to address the
content of the corresponding cell.
In the spreadsheet cells you can enter not only numbers, but all types of mathematical
objects that are supported by GeoGebra (e. g., coordinates of points, functions, commands).
If possible, GeoGebra immediately displays the graphical representation of the object you
entered in a spreadsheet cell in the Graphics View as well. Thereby, the name of the object
matches the name of the spreadsheet cell used to initially create it (e. g., A5, C1).
Note: By default, spreadsheet objects are classified as auxiliary objects in the Algebra View.
You can show or hide these auxiliary objects by selecting Auxiliary Objects from the View
menu.
1.2. GeoGebra as a Tool for Teaching and Learning Mathematics
1.2.1. Customizing the User Interface
The user interface of GeoGebra can be customized by using the View menu. For example,
you can hide different parts of the interface (e. g., the Algebra View, Spreadsheet View, or
Input Bar) by checking or un‐checking the corresponding menu item in the View menu.
Showing and Hiding Objects
You may show or hide objects in the Graphics View in different ways.
• You may use tool Show/Hide Object to show or hide objects.
• Open the Context Menu and select item Show Object to change the visibility
status of the selected object.
• In the Algebra View, the icon to the left of every object shows its current visibility
state ( shown or hidden). You may directly click on the little marble icon in
order to change the visibility status of an object.
• You can also use the tool Check Box to Show/Hide Objects in order to show or
hide one or several objects.
Customizing the Graphics View
In order to adjust the visible part of the drawing pad in the Graphics View, you can drag the
drawing pad by using tool Move Drawing Pad and use the following ways of zooming:
• You may use the tools Zoom In and Zoom Out in order to zoom in the Graphics
View.
Note: The position of your click determines the center of zoom.
• You may use the scroll wheel of your mouse in order to zoom in the Graphics View.
8
• You may use keyboard shortcuts to zoom in (Ctrl +) and to zoom out (Ctrl ‐).
• After right clicking (Mac OS: Ctrl‐click) on an empty spot on the drawing pad a
Context Menu appears which allows you to Zoom.
• You may specify a zoom rectangle by right clicking (Mac OS: Cmd ‐ click) on an empty
spot in the Graphics View and dragging the mouse to the opposite corner of your
desired zoom rectangle. Release the mouse button in order to finish the zoom
rectangle, which will then automatically adjust to fill all the space in the Graphics
View.
You can also show or hide the coordinate axes and a coordinate grid in the Graphics View by
using the View menu.
Note: Another way of showing or hiding the axes and the grid is by right clicking (Mac OS:
Ctrl‐click) on the drawing pad and selecting the corresponding items Axes or Grid from
the appearing Context Menu.
Customizing Coordinate Axes and Grid
The coordinate axes and grid can be customized using the Properties Dialog of the Graphics
View. After right clicking (Mac OS: Ctrl‐click) on the drawing pad, you can open this dialog
window by selecting Properties from the appearing Context Menu of the Graphics View.
• On tab Axes, you can, for example, change the line style and units of the coordinate
axes, and set the distance of the tickmarks to a certain value. Note that you can
customize both axes individually, by clicking on tabs xAxis or yAxis. Furthermore, you
can also change the ratio between the axes and hide or show the axes individually.
• On tab Grid, you can, for example, change the color and line style of the coordinate
grid, and set the distance for grid lines to a certain value. In addition, you may also
set the grid to be Isometric.
Note: Scaling the axes is possible in every mode by pressing and holding the Shift‐key (PC:
also Ctrl‐key) while dragging the axis.
Note: The Properties Dialog of the Graphics View is different from the Properties Dialog for
objects.
Customizing the Toolbar
The Toolbar can be customized by selecting Customize Toolbar… from the Tools menu.
Select the tool or toolbox you want to remove from the Toolbar in the list on the left hand
side of the appearing dialog window and click button Remove > in order to remove the
tool/toolbox from the Toolbar.
Note: You can restore the default Toolbar by clicking on the button Restore Default Toolbar
in the left lower corner of the dialog window.
9
1.2.2. The Properties Dialog
The Properties Dialog allows you to modify properties of objects (e. g., size, color, filling, line
style, line thickness, visibility).
You can open the Properties Dialog in several ways:
• Right click (Mac OS: Ctrl‐click) on an object and select Properties… from the
appearing Context Menu.
• Select item Properties from the Edit menu.
• Select the Move tool and double click on an object in the Graphics View. In the
appearing Redefine dialog window, click on the button Properties….
In the Properties Dialog objects are organized by types (e. g., points, lines, circles) in the list
on the left hand side, which makes it easier to handle large numbers of objects. You need to
select one or more objects from this list in order to change its/their properties.
Note: By clicking on a heading in the list of objects (e. g., Point) you can select all objects of
this type and therefore, quickly change the properties for all these objects.
You can modify the properties of selected objects using the tabs on the right hand side (e.
g., Basic, Color, Style, Advanced).
Note: Depending on the selection of objects in the list, a different set of tabs may be
available.
Close the Properties Dialog when you are done with changing properties of objects.
1.2.3. The Context Menu
The Context Menu provides a quick way to change the behavior or advanced properties of
an object. Right click (Mac OS: Ctrl‐click) on an object in order to open its Context Menu. For
example, it allows you to change the object's algebraic notation (e. g., polar or Cartesian
coordinates, implicit or explicit equation) and to directly access features like Rename,
Delete, Trace On, Animation On, or Copy to Input Bar.
Note: If you open the Context Menu for a point in the Graphics View, it gives you the option
Trace to Spreadsheet (only if the Spreadsheet View is active). Once selected, this feature
allows you to record the coordinates of the point in the Spreadsheet View if it is moved.
Note: Selecting Properties… in the Context Menu opens the Properties Dialog, where you
can change the properties of all objects used.
10
1.3. GeoGebra as a Presentation Tool
1.3.1. The Navigation Bar
GeoGebra offers a Navigation Bar that allows you to navigate through the construction
steps of a prepared GeoGebra file. Select item Navigation Bar for Construction Steps in the
View menu in order to display the Navigation Bar at the bottom of the Graphics View.
The Navigation Bar provides a set of navigation buttons and displays the number of
construction steps (e. g., 2 / 7 means that currently the second step of a total of 7
construction steps is displayed):
• button: go back to step 1
• button: go back step by step
• button: go forward step by step
• button: go to the last step
• Play: automatically play the construction step by step
Note: You may change the speed of this automatic play feature using the text box to
the right of the Play button.
• Pause: pause the automatic play feature
Note: This button only appears after you click on the Play button.
• button: This button opens the Construction Protocol.
1.3.2. The Construction Protocol
You can access the interactive Construction Protocol by selecting item Construction Protocol
from the View menu. It is a table that shows all construction steps. The Construction
Protocol allows you to redo a prepared construction step by step using the Navigation Bar
at the bottom of the Construction Protocol dialog.
Navigating and Modifying the Construction Protocol
You may use the keyboard to navigate in the Construction Protocol:
• Use the ↑ up arrow of your keyboard to go to the previous construction step.
• Use the ↓ down arrow of you keyboard to go to the next construction step.
• Use the Home key to go to the beginning of the Construction Protocol.
• Use the End key to go to the end of the Construction Protocol.
• Use the Delete key in order to delete the selected construction step.
Note: This may also affect other objects that depend on the selected
object/construction step.
You may also use the mouse in order to navigate in the Construction Protocol:
• Double click a row in order to select a construction step.
• Double click the header of any column in order to go to the start of the Construction
Protocol.
11
• Drag and drop a row in order to move a construction step to another position in the
Construction Protocol.
Note: This is not always possible due to the dependencies between different objects.
• Right click a row in order to open the Context Menu for the object of this
construction step.
Note: You can insert construction steps at any position. Select the construction step below
you would like to insert a new construction step. Leave the Construction Protocol window
open while you create a new object. This new construction step is immediately inserted into
the selected position of the Construction Protocol.
Using the column Breakpoint in the View menu of the Construction Protocol window, you
are able to define certain construction steps as Breakpoints. This allows you to group several
objects together. When navigating through your construction using the Navigation Bar,
groups of objects are shown at the same time.
Note: You may switch the different columns of the Construction Protocol on and off by
using the View menu of the Construction Protocol window.
Exporting the Construction Protocol as a Webpage
GeoGebra allows you to export the Construction Protocol as a webpage. First, you need to
open the Construction Protocol using the View menu. Then, you can open the File menu of
the appearing Construction Protocol window and select item Export as Webpage.
In the export window of the Construction Protocol you can enter Title, Author, and a Date
for the construction and choose whether or not you want to include a picture of the
Graphics View and the Algebra View. In addition, you can also choose to export a Colorful
Construction Protocol. This means that objects in the Construction Protocol will match the
color of the corresponding objects in the construction.
Note: The exported HTML file can be viewed with any Internet browser (e. g. Firefox,
Internet Explorer) and edited with many text processing systems (e. g. OpenOffice Writer).
1.3.3. Customize the Settings
GeoGebra allows you to change and save settings using the Options menu. For example, you
may change the Angle Unit from Degree to Radians, or change the Point Style, Checkbox
Size, and Right Angle Style. In addition, you may change how Coordinates are displayed on
screen and which objects are labeled (Labeling).
Please see the section about the Options menu for more information.
You can save your customized settings by selecting item Save Settings from the Options
menu. After doing so, GeoGebra will remember your customized settings and use them for
every new GeoGebra file you create.
Note: You may restore the default settings by selecting Restore Default Settings from the
Options menu.
12
Note: If you use GeoGebra as a presentation tool, you might want to increase the Font Size
(Options menu) so your audience can easily read text and labels of objects.
1.4. GeoGebra as an Authoring Tool
1.4.1. Printing Options
Printing the Graphics View
GeoGebra allows you to print the Graphics View of your constructions. You can find the
corresponding item Print Preview in the File menu. In the appearing Print Preview dialog
window, you can specify the Title, Author, and a Date for the construction. In addition, you
can set the Scale of your printout (in cm) and change the Orientation of the paper used
(portrait or landscape).
Note: In order to update the Print Preview after you made changes to the text or layout of
the printout, you need to press the Enter‐key.
Printing the Construction Protocol
If you want to print the Construction Protocol, you first need to open the Construction
Protocol dialog window by using the View menu. Then, you can open the Print Preview
window of the Construction Protocol from the File menu of this new window.
Again, you may enter Title, Author, and a Date or change the Scale or paper Orientation
before printing your Construction Protocol.
Note: You may switch the different columns Name, Definition, Command, Algebra, and
Breakpoint of the Construction Protocol on and off by using the View menu of the
Construction Protocol dialog window.
1.4.2. Creating Pictures of the Graphics View
Saving the Graphics View as a Picture
You can save the Graphics View of your constructions as a picture file on your computer.
Note: The full Graphics View will be saved as a picture. If your construction does not use all
the available space in the Graphics View, you might want to…
• …use tools Move Drawing Pad, Zoom In, and/or Zoom Out in order to place
your construction in the upper left corner of the Graphics View. Afterwards, you may
reduce the size of the GeoGebra window by dragging one of its corners with the
mouse.
• … use the selection rectangle in order to specify which part of the Graphics View
should be exported and saved as a picture.
13
• You may create points called Export_1 and Export_2, which will be used to define
diagonally opposite corners of the export rectangle.
Note: Points Export1 and Export2 must be within the visible area of the Graphics
View.
In the File menu, select item Export before clicking on item Graphics View as Picture. In
the appearing dialog window you may specify the Format, Scale (in cm), and the Resolution
(in dpi) of the output picture file.
Note: The true size of the exported image is shown at the bottom of the export window just
above the buttons, both in centimeters and pixel.
Please find more information about the different picture files available in section Export ‐
Graphics View as Picture (png, eps)….
Copying the Graphics View to Clipboard
There are different ways of copying the Graphics View to your computer's clipboard:
• In the Edit menu, you may select item Graphics View to Clipboard.
• In the File menu, you first need to select item Export, before you can click on item
Graphics View to Clipboard.
• In the Export Graphics View as Picture dialog window (menu File – Export –
Graphics View as Picture (png, eps)…) you may click on the button Clipboard.
This feature copies a screenshot of the Graphics View to your system's clipboard as a PNG
(see PNG format) picture. This picture can be pasted into other documents (e. g. a word
processing document).
Note: In Order to export your construction at a certain scale (in cm) please use the menu
item Graphics View as Picture in the File menu, Export.
1.4.3. Creating Interactive Webpages
GeoGebra allows you to create interactive webpages, so called Dynamic Worksheets, from
your files. In the File menu, you need to select item Export before you can click on item
Dynamic Worksheet as Webpage (html). This opens the export dialog window for
Dynamic Worksheets:
• At the top of the export window you can enter the Title, Author, and a Date for your
Dynamic Worksheet.
• Tab General allows you to add some text above and below the dynamic construction
(e. g., a description of the construction and some tasks). You can also determine if
the construction itself may be included directly into the webpage or if it can be
opened by clicking on a button.
• Tab Advanced allows you to change the functionality of the dynamic construction (e.
g., show a reset icon, double click should open the GeoGebra application window) as
well as to modify the user interface shown in the interactive applet (e. g., show the
Toolbar, modify height and width).
14
Note: If the size of your applet is too big to fit on a computer screen with standard
resolution (1024 x 768), you may want to resize it before the actual export as a
Dynamic Worksheet.
Note: Several files are created when you export a Dynamic Worksheet:
• HTML file (e. g. circle.html) – this file includes the worksheet itself
• GGB file (e. g. circle.ggb) – this file includes your GeoGebra construction
• JAR (several files) – these files include GeoGebra and make your worksheet
interactive
All these files (e. g. circle.html, circle.ggb and the geogebra.jar files) have to be in one folder
(directory) to let the dynamic construction work.
The exported HTML file (e. g. circle.html) can be viewed with any Internet browser (e. g.
Mozilla, Internet Explorer, Safari). In order to let the dynamic construction work, Java has to
be installed on the computer. You can get Java from without charge. If
you want to use your Dynamic Worksheet in your school's computer network, ask your local
network administrator to install Java on the computers.
Note: You can edit the Dynamic Worksheet's text with many word processing systems (e. g.
FrontPage, OpenOffice Writer) by opening the exported HTML file. You may also edit the
Dynamic Worksheet's applet by opening the GGB file in GeoGebra and saving it with the
same name afterwards.
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2. Geometric Input
2.1. General Notes
The Graphics View shows the graphical representation of mathematical objects (e. g.,
points, vectors, segments, polygons, functions, curves, straight lines, conic sections).
Whenever the mouse is moved over one of these objects a description appears as a roll‐
over text and the object is highlighted.
There are several tools/modes to tell GeoGebra how it should react to mouse input in the
Graphics View (see section Construction Tools). For example, clicking on the drawing pad
can create a new point (see tool New Point), intersect two objects (see tool Intersect
Two Objects), or create a circle (see Circle tools).
2.2. Construction Tools
The following construction tools or modes can be activated by clicking on the buttons of the
Toolbar. You can click on the small arrow in the lower right corner of an icon to open a
Toolbox with similar other tools.
Note: With most construction tools you can easily create new points by clicking on empty
spaces on the drawing pad.
Selecting Objects
To select an object means to click on it with the mouse after selecting the Move tool.
If you want to select several objects at the same time, you could draw a selection rectangle:
Select the Move tool and click on the position of the first corner of your desired selection
rectangle. Hold the left mouse key pressed down and move the pointer to the position of
the diagonally opposite corner of your desired selection rectangle. After releasing the mouse
button, all objects within the selection rectangle are selected.
Note: By holding the Ctrl‐key (Mac OS: Cmd‐key) while clicking on different objects, you can
select several objects at the same time.
Fast Renaming of Objects
To quickly rename a selected or newly created object just start typing to open the Rename
dialog for this object. Then, type in the new name of the selected object and click on the OK
button.
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2.2.1. General Tools
Copy Visual Style
This tool allows you to copy visual properties (e. g., color, size, line style) from one object to
one or more other objects. To do so, first select the object whose properties you want to
copy. Then, click on all other objects that should adopt these properties.
Delete Object
Click on any object you want to delete (also see command Delete).
Note: You can use the Undo button if you accidentally deleted the wrong object.
Move
Drag and drop free objects with the mouse. If you select an object by clicking on it in
Move mode, you may…
• … delete the object by pressing the Delete‐key
• … move the object by using the arrow keys (see section Manual Animation)
Note: You can quickly activate the Move tool by pressing the Esc‐key of your keyboard.
Move Drawing Pad
Drag and drop the drawing pad in the Graphics View to change its visible area.
Note:
• You can also move the drawing pad by pressing the Shift‐key (MS Windows: also Ctrl‐
key) and dragging it with the mouse in any mode.
• In this mode you can also scale each of the axes by dragging it with the mouse.
Record to Spreadsheet
This tool allows you to move an object and to record a sequence of its values in the
Spreadsheet View. This tool works for numbers, points, and vectors.
Note: GeoGebra will use the first two empty columns of the Spreadsheet View to record the
values of the selected objects.
Relation
Select two objects to get information about their relation in a pop‐up window (also see
command Relation).
17
Rotate around Point
Select the center point of the rotation first. Then, you may rotate free objects around this
point by dragging them with the mouse (also see command Rotate).
Show / Hide Label
Click on an object to show or hide its label.
Show/Hide Object
Select the object you want to show or hide after activating this tool. Then, switch to another
tool in order to apply the visibility changes to this object.
Note: When you activate this tool, all objects that should be hidden are displayed in the
Graphics View highlighted. In this way, you can easily show hidden objects again by
deselecting them before switching to another tool.
Zoom In
Click on any place on the drawing pad to zoom in (also see section Customizing the Graphics
View).
Note: The position of your click determines the center of zoom.
Zoom Out
Click on any place on the drawing pad to zoom out (also see section Customizing the
Graphics View).
Note: The position of your click determines the center of zoom.
2.2.2. Point Tools
Intersect Two Objects
Intersection points of two objects can be created in two ways (also see command Intersect).
• Selecting two objects creates all intersection points (if possible).
• Directly clicking on an intersection of the two objects creates only this single
intersection point .
Note: For segments, rays, or arcs you may specify whether you want to Allow outlying
intersections on tab Basic of the Properties Dialog. This can be used to get intersection
points that lie on the extension of an object. For example, the extension of a segment or a
ray is a straight line.
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Midpoint or Center
You may click on either two points or one segment to get its midpoint. You can also click on
a conic section (circle or ellipse) in order to create its center point (also see commands
Center and Midpoint).
New Point
Click on the drawing pad in the Graphics View in order to create a new point. The
coordinates of the point are fixed when the mouse button is released.
Note:
• By clicking on a segment, straight line, polygon, conic section, function, or curve you
can create a point on this object (also see command Point).
• Clicking on the intersection of two objects creates this intersection point (also see
tool Intersect Two Objects and command Intersect).
2.2.3. Vector Tools
Vector between Two Points
Select the starting point and then the end point of the vector (also see command Vector).
Vector from Point
Select a point A and a vector v to create the new point B = A + v as well as the vector from A
to B (also see command Vector).
2.2.4. Segment Tools
Segment between Two Points
Select two points A and B in order to create a segment between A and B (also see command
Segment).
Note: In the Algebra View, the segment's length is displayed.
Segment with Given Length from Point
Click on a point A that should be the starting point of the segment. Specify the desired
length a of the segment in the appearing window (also see command Segment).
Note: This tool creates a segment with length a and endpoint B which may be rotated
around the starting point A by using tool Move.
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2.2.5. Ray Tool
Ray through Two Points
Selecting two points A and B creates a ray starting at A through B (also see command Ray).
Note: In the Algebra View the equation of the corresponding line is displayed.
2.2.6. Polygon Tools
Polygon
Successively select at least three points which will be the vertices of the polygon. Then, click
the first point again in order to close the polygon (also see command Polygon).
Note: In the Algebra View, the polygon's area is displayed.
Regular Polygon
Select two points A and B and specify the number n of vertices in the text field of the
appearing dialog window. This gives you a regular polygon with n vertices including points A
and B (also see command Polygon).
2.2.7. Line Tools
Angle Bisector
Angle bisectors can be defined in two ways (also see command AngleBisector):
• Selecting three points A, B, and C produces the angle bisector of the enclosed angle,
where point B is the apex.
• Selecting two lines produces their two angle bisectors.
Note: The direction vectors of all angle bisectors have length 1.
Best Fit Line
Create the best fit line for a set of points in the following ways (also see command FitLine):
• Create a selection rectangle that contains all points.
• Select a list of points to create their corresponding best fit line.
Line through Two Points
Selecting two points A and B creates a straight line through A and B (also see command
Line).
Note: The line's direction vector is (B ‐ A).
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Parallel Line
Selecting a line g and a point A defines a straight line through A parallel to g (also see
command Line).
Note: The line's direction is the direction of line g.
Perpendicular Bisector
Click on either a segment s or two points A and B in order to create a perpendicular bisector
(also see command PerpendicularBisector).
Note: The bisector's direction is equivalent to the perpendicular vector of segment s or AB
(also see command PerpendicularVector).
Perpendicular Line
Selecting a line g and a point A creates a straight line through A perpendicular to line g (also
see command PerpendicularLine).
Note: The line's direction is equivalent to the perpendicular vector of g (also see command
PerpendicularVector).
Polar or Diameter Line
This tool creates the polar or diameter line of a conic section (also see command Polar).
• Select a point and a conic section to get the polar line.
• Select a line or a vector and a conic section to get the diameter line.
Tangents
Tangents to a conic section can be produced in several ways (also see command Tangent):
• Selecting a point A and a conic c produces all tangents through A to c.
• Selecting a line g and a conic c produces all tangents to c that are parallel to line g.
• Selecting a point A and a function f produces the tangent line to f in x = x(A).
Note: x(A) represents the x‐coordinate of point A. If point A lies on the function
graph, the tangent runs through point A.
2.2.8. Conic Section Tools
Circle with Center and Radius
Select the center point M and enter the radius in the text field of the appearing dialog
window (also see command Circle).
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Circle with Center through Point
Selecting a point M and a point P defines a circle with center M through P (also see
command Circle).
Note: The circle's radius is the distance MP.
Circle through Three Points
Selecting three points A, B, and C defines a circle through these points (also see command
Circle).
Note: If the three points lie on one straight line, the circle degenerates to this line.
Compass
UK English: Compasses
Select a segment or two points to specify the radius. Then, click on a point that should be
the center of the new circle.
Conic through Five Points
Selecting five points produces a conic section through these points (also see command
Conic).
Note: If four of these five points lie on a line, the conic section is not defined.
Ellipse
Select the two foci of the ellipse. Then, specify a third point that lies on the ellipse (also see
command Ellipse).
Hyperbola
Select the two foci of the hyperbola. Then, specify a third point that lies on the hyperbola
(also see command Hyperbola).
Parabola
Select a point and the directrix of the parabola (also see command Parabola).
2.2.9. Arc and Sector Tools
Note: In GeoGebra, the algebraic value of an arc is its length. The value of a sector is its area.
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Circular Arc with Center between Two Points
First, select the center point M of the circular arc. Then, select the starting point A of the
arc, before you select a point B that specifies the length of the arc (also see command
CircularArc).
Note: While point A always lies on the circular arc, point B does not have to lie on it.
Circular Sector with Center between Two Points
First, select the center point M of the circular sector. Then, select the starting point A of the
sector's arc, before you select a point B that specifies the length of the sector's arc (also see
command CircularSector).
Note: While point A always lies on the sector's arc, point B does not have to lie on it.
Circumcircular Arc through Three Points
Selecting three points A, B, and C creates a circular arc through these points. Thereby, point
A is the starting point of the arc, point B lies on the arc, and point C is the endpoint of the
arc (also see command CircumcircularArc).
Circumcircular Sector through Three Points
Selecting three points A, B, and C creates a circular sector through these points. Thereby,
point A is the starting point of the sector's arc, point B lies on the arc, and point C is the
endpoint of the sector's arc (also see command CircumcircularSector).
Semicircle
Select two points A and B to create a semicircle above the segment AB (also see command
Semicircle).
2.2.10. Number and Angle Tools
Angle
With this tool you can create angles in different ways (also see command Angle):
• Click on three points to create an angle between these points. The second point
selected is the vertex of the angle.
• Click on two segments to create the angle between them.
• Click on two lines to create the angle between them.
• Click on two vectors to create the angle between them.
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• Click on a polygon to create all angles of this polygon.
Note: If the polygon was created by selecting its vertices in counter clockwise
orientation, the Angle tool gives you the interior angles of the polygon.
Note: Angles are created in counter clockwise orientation. Therefore, the order of selecting
these objects is relevant for the Angle tool. If you want to limit the maximum size of an
angle to 180°, un‐check Allow Reflex Angle on tab Basic of the Properties Dialog.
Angle with Given Size
Select two points A and B and type the angle's size into the text field of the appearing
window (also see command Angle).
Note: This tool creates a point C and an angle α, where α is the angle ABC.
Area
This tool gives you the area of a polygon, circle, or ellipse as a number and shows a dynamic
text in the Graphics View (also see command Area).
Distance or Length
This tool gives you the distance between two points, two lines, or a point and a line as a
number and shows a dynamic text in the Graphics View. It can also give you the length of a
segment, the circumference of a circle, or the perimeter of a polygon (also see commands
Distance and Length).
Slider
Click on any free place in the Graphics View to create a slider for a number or an angle. The
appearing dialog window allows you to specify the Name, Interval [min, max], and
Increment of the number or angle, as well as the Alignment and Width of the slider (in pixel).
Note: In the Slider dialog window you can enter a degree symbol ° or pi (π) for the interval
and increment by using the following keyboard shortcuts:
• Alt‐O (Mac OS: Ctrl‐O) for the degree symbol °
• Alt‐P (Mac OS: Ctrl‐P) for the pi symbol π
The position of a slider may be absolute in the Graphics View (this means that the slider is
not affected by zooming, but always remains in the visible part of the Graphics View) or
relative to the coordinate system (see Properties Dialog of the corresponding number or
angle).
Note: In GeoGebra, a slider is the graphical representation of a free number or free angle.
You can easily create a slider for any existing free number or angle by showing this object in
the Graphics View (see Context Menu; see tool Show/Hide Object).
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Slope
This tool gives you the slope of a line and shows a slope triangle in the Graphics View (also
see command Slope).
2.2.11. Boolean Variable Tool
Check Box to Show/Hide Objects
Clicking in the Graphics View creates a check box (see section Boolean Variables and
Operations) that allows you to show and hide one or more objects. In the appearing dialog
window you can specify which objects should be affected by the check box.
Note: You may select these objects from the list provided in the dialog window or select
them with the mouse in any view.
2.2.12. Locus Tool
Locus
Select a point B that depends on another point A and whose locus should be drawn. Then,
click on point A to create the locus of point B (also see command Locus).
Note: Point A has to be a point on an object (e. g. line, segment, circle).
Example:
• Type f(x) = x^2 – 2 x – 1 into the Input Bar and press the Enter‐key.
• Place a new point A on the x‐axis (see tool New Point; see command Point).
• Create point B = (x(A), f'(x(A))) that depends on point A.
• Select tool Locus and successively click on point B and point A.
• Drag point A along the x‐axis to see point B moving along its locus line.
2.2.13. Geometric Transformation Tools
The following geometric transformations work for points, lines, conic sections, polygons,
and images.
Dilate Object from Point by Factor
UK English: Enlarge Object from Point by Factor
Select the object to be dilated. Then, click on a point to specify the dilation center and enter
the dilation factor into the text field of the appearing dialog window (also see commands
Dilate (US) and Enlarge (UK)).
25
Reflect Object about Line
UK English: Reflect Object in Line
Select the object you want to reflect. Then, click on a line to specify the mirror/line of
reflection (also see command Reflect).
Reflect Object about Point
UK English: Reflect Object in Point
Select the object you want to reflect. Then, click on a point to specify the mirror/point of
reflection (also see command Reflect).
Reflect Point about Circle
UK English: Reflect Point in Circle
This tool allows you to invert a point in a circle. Select the point you want to invert. Then,
click on a circle to specify the mirror/circle of inversion (also see command Reflect).
Rotate Object around Point by Angle
Select the object you want to rotate. Then, click on a point to specify the center of rotation
and enter the rotation angle into the text field of the appearing dialog window (also see
command Rotate).
Translate Object by Vector
Select the object you want to translate. Then, click on the translation vector (also see
command Translate).
2.2.14. Text Tool
Insert Text
With this tool you can create static and dynamic text or LaTeX formulas in the Graphics View
(also see section Text Commands).
At first, you need to specify the location of the text in one of the following ways:
• Click in the Graphics View to create a new text at this location.
• Click on a point to create a new text that is attached to this point.
Then, a dialog appears where you may enter your text.
26
Note: You may specify the position of a text as absolute on screen or relative to the
coordinate system on tab Basic of the Properties Dialog.
Static text does not depend on any mathematical objects and is usually not affected by
changes of the construction.
Dynamic text contains values of objects that automatically adapt to changes made to these
objects.
Mixed text is a combination of static and dynamic text. In order to create a mixed text you
may enter the static part of the text using the keyboard (e. g., Point A =). Then, click on
the object whose value you want to display in the dynamic part of the text.
Note: GeoGebra automatically adds the syntax ("Point A = " + A ) necessary to create
your mixed text: quotation marks around the static part of the text and a plus (+) symbol to
connect the different parts of the text.
Input Description
This is static text Static text
A Dynamic text (if point A exists)
"Point A = " + A Two‐part mixed text using the value of point A
Three‐part mixed text using the value of
"a = " + a + "cm"
number a
Note: If an object with the name xx already exists and you want to create a static text using
the object's name, you need to enter it with quotation marks ("xx"). Otherwise, GeoGebra
will automatically create a dynamic text that gives you the value of object xx instead of its
name. However, you can type any text that doesn't match any existing object's name
without the quotation marks.
Note: Within a mixed text, the static part needs to be in between a pair of quotation marks.
Different parts of a text (e. g., static and dynamic parts) need to be connected using plus (+)
symbols.
LaTeX Formulas
In GeoGebra you can write formulas as well. To do so, check the box LaTeX formula in the
dialog window of the Insert Text tool and enter your formula in LaTeX syntax.
Note: In order to create text that contains a LaTeX formula as well as static text you may
enter the static part of the text and then add the LaTeX formula in between a set of dollar
symbols ($).
Example: The length of the diagonal is $sqrt{ 2 }$.
You can select the syntax for common formula symbols from the drop‐down menu next to
the LaTeX checkbox. This inserts the corresponding LaTeX code into the text field and places
the cursor in between a set of curly brackets. If you would like to create dynamic text within
the formula, you need to click on an object causing GeoGebra to insert its name as well as
the syntax for mixed text.
27
Some important LaTeX commands are explained in following table. Please have a look at any
LaTeX documentation for further information.
LaTeX input Result
a cdot b a ⋅b
a
frac{a}{b}
b
sqrt{x} x
sqrt[n]{x} n
x
r
vec{v} v
overline{AB} AB
x^{2} x2
a_{1} a1
sinalpha +
sin α + cos β
cosbeta
b
int_{a}^{b} x dx ∫ xdx
a
∑ i
n 2
sum_{i=1}^{n} i^2 i =1
2.2.15. Image Tool
Insert Image
This tool allows you to insert an image into the Graphics View.
First, specify the location of the image in one of the following two ways:
• Click in the Graphics View to specify the position of the image's lower left corner.
• Click on a point to specify this point as the lower left corner of the image.
Then, a file‐open dialog appears that allows you to select the image file from the files saved
on your computer.
Note: After selecting the tool Insert Image, you can use the keyboard shortcut Alt‐click in
order to paste an image directly from your computer's clipboard into the Graphics View.
Properties of Images
The position of an image may be absolute on screen or relative to the coordinate system.
You can specify this on tab Basic of the Properties Dialog of the image.
You may specify up to three corner points of the image on tab Position of the Properties
Dialog. This gives you the flexibility to scale, rotate, and even distort images (also see
command Corner).
• Corner 1: position of the lower left corner of the image
28
• Corner 2: position of the lower right corner of the image
Note: This corner may only be set if Corner 1 was set before. It controls the width of
the image.
• Corner 4: position of the upper left corner of the image
Note: This corner may only be set if Corner 1 was set before. It controls the height of
the image.
Example: Create three points A, B, and C to explore the effects of the corner points.
• Set point A as the first and point B as the second corner of your image. By dragging
points A and B in Move mode you can explore their influence.
• Now, remove point B as the second corner of the image. Set point A as the first and
point C as the fourth corner and explore how dragging the points now influences the
image.
• Finally, you may set all three corner points and see how dragging the points distorts
your image.
Example: You already saw how to influence the position and size of your image. If you want
to attach your image to a point A and set its width to 3 and its height to 4 units, you could
do the following:
• Set Corner 1 to A
• Set Corner 2 to A + (3, 0)
• Set Corner 4 to A + (0, 4)
Note: If you now drag point A in Move mode, the size of your image does not change.
You may specify an image as a Background Image on tab Basic of the Properties Dialog. A
background image lies behind the coordinate axes and cannot be selected with the mouse
any more.
Note: In order to change the background setting of an image, you may open the Properties
Dialog by selecting Properties… from the Edit menu.
The Transparency of an image can be changed in order to see objects or axes that lie behind
the image. You can set the transparency of an image by specifying a Filling value between 0
% and 100 % on tab Style of the Properties Dialog.
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3. Algebraic Input
3.1. General Notes
The algebraic representations of mathematical objects (e. g., values, coordinates, equations)
are shown in the Algebra View. You can create and modify objects by using the Input Bar at
the bottom of the GeoGebra window (see sections Direct Inputand and Commands).
Note: Always press the Enter‐key after typing algebraic input into the Input Bar.
Note: Pressing the Enter‐key at any time toggles the focus between the Input Bar and the
Graphics View. This allows you to enter expressions and commands into the Input Bar
without having to click on it with the mouse first.
Naming Objects
You can assign a certain name to an object when you create it using the Input Bar:
• Points: In GeoGebra, points are always named using upper case letters. Just type in
the name (e. g., A, P) and an equal sign in front of the coordinates or commands.
Examples: C = (2, 4), P = (1; 180°), Complex = 2 + i
• Vectors: In order to distinguish between points and vectors, vectors need to have a
lower case name in GeoGebra. Again, type in the name (e. g., v, u) and an equal sign
in front of the coordinates or commands.
Examples: v = (1, 3), u = (3; 90°), complex = 1 – 2i
• Lines, circles, and conic sections: These objects can be named by typing in the name
and a colon in front of their equations or commands.
Examples: g: y = x + 3, c: (x-1)^2 + (y – 2)^2 = 4,
hyp: x^2 – y^2 = 2
• Functions: You can name functions by typing, for example, f(x) = or g(x)= in front
of the function's equation or commands.
Examples: h(x) = 2 x + 4, q(x) = x^2, trig(x) = sin(x)
Note:
• If you don't manually assign a name to an object, GeoGebra assigns the names of
new objects in alphabetical order.
• You can create indices within the names of objects by using an underscore. For
example A1 is entered as A_1 and sAB is entered as s_{AB}.
Change Values
There are two ways of manipulating a free object's value:
• Change the value of the object by entering its name and the new value in the Input
Bar (see section Direct Input).
30
Example: If you want to change the value of an existing number a = 3, type
a = 5 into the Input Bar and press the Enter‐key.
• Edit the algebraic representation: Activate tool Move and double click on the
object in the Algebra View. This opens a text box where you can edit the object's
value. Press the Enter‐key to apply your changes.
Note: While free objects' values can be changed directly, the values of dependent objects
can only be influenced by changing their parent objects or by redefining the dependent
object.
Display Input Bar History
After placing the cursor in the Input Bar you can use the ↑ up and ↓ down arrow keys of
your keyboard in order to navigate through prior input step by step.
Note: Click on the little question mark to the left of the Input Bar in order to display the
help feature for the Input Bar.
Insert Name, Value, or Definition of an Object into the Input Bar
Insert the name of an object: Activate tool Move and select the object whose name you
want to insert into the Input Bar. Then, press the F5 key on your keyboard.
Note: The name of the object is appended to any expression you typed into the Input Bar
before pressing the F5 key.
Insert the value of an object: There are two ways of inserting an object's value (e. g., (1, 3),
3x – 5y = 12) into the Input Bar.
• Right click (Mac OS: Ctrl‐click) on the object and select item Copy to Input Bar
from the appearing Context Menu.
• Activate tool Move and select the object whose value you want to insert into the
Input Bar. Then, press the F4 key on your keyboard.
Note: The value of the object is appended to any expression you typed into the Input
Bar before pressing the F4 key.
Insert the definition of an object: There are two ways of inserting an object's definition (e.
g., A = (4, 2), c = Circle[A, B]) into the Input Bar.
• Alt click on the object to insert the object's definition and delete whatever input
might have been in the Input Bar before.
• Activate tool Move and select the object whose definition you want to insert into
the Input Bar. Then, press the F3 key on your keyboard.
Note: The definition of the object replaces any expression you typed into the Input
Bar before pressing the F3 key.
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3.2. Direct Input
GeoGebra can work with numbers, angles, points, vectors, segments, lines, conic sections,
functions, and parametric curves. You can enter these objects into the Input Bar by using
their coordinates or equations and pressing the Enter‐key.
3.2.1. Numbers and Angles
Numbers
You can create numbers by using the Input Bar. If you only type in a number (e. g., 3),
GeoGebra assigns a lower case letter as the name of the number. If you want to give your
number a specific name, you can type in the name followed by an equal sign and the
number (e. g., create a decimal r by typing in r = 5.32).
Note: In GeoGebra, numbers and angles use a period (.) as a decimal point.
You can also use the constant π and the Euler constant e for expressions and calculations by
selecting them from the drop down menu next to the Input Bar or by using keyboard
shortcuts.
Note: If the variable e is not used as a name of an existing object yet, GeoGebra will
recognize it as the Euler constant if you use it in new expressions.
Angles
Angles are entered in degree (°) or radians (rad). The constant π is useful for radian values
and can also be entered as pi.
Note: You can enter a degree symbol (°) or the pi symbol (π) by using the following keyboard
shortcuts:
• Alt‐O (Mac OS: Ctrl‐O) for the degree symbol °
• Alt‐P (Mac OS: Ctrl‐P) for the pi symbol π
Example: You can enter an angle α in degree (e. g., α = 60°) or in radians (e. g.,
α = pi/3).
Note: GeoGebra does all internal calculations in radians. The degree symbol (°) is nothing
but the constant π/180 used to convert degree into radians.
Examples:
• If a = 30 is a number, then α = a° converts number a to an angle α = 30°, without
changing its value.
• If you type in b = α / °, the angle α is converted back to the number b = 30,
without changing its value.
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Sliders and Arrow Keys
Free numbers and angles can be displayed as sliders in the Graphics View (see tool
Slider). Using the arrow keys, you may change the value of numbers and angles in the
Algebra View too (see section Manual Animation).
Limit Value to Interval
Free numbers and angles may be limited to an interval [min, max] by using tab Slider of the
Properties Dialog (see also tool Slider).
Note: For dependent angles you can specify whether they may become reflex or not on tab
Basic of the Properties Dialog.
3.2.2. Points and Vectors
Points and vectors may be entered in Cartesian or polar coordinates (see section Numbers
and Angles).
Note: Upper case labels denote points whereas lower case labels refer to vectors.
Examples:
• To enter a point P or a vector v in Cartesian coordinates you may use P = (1, 0)
or v = (0, 5).
• In order to use polar coordinates type in P = (1; 0°) or v = (5; 90°).
Note: You need to use a semicolon to separate the two coordinates. If you don't type
in the degree symbol, GeoGebra will treat the angle as if entered in radians.
In GeoGebra, you can also do calculations with points and vectors.
Examples:
• You can create the midpoint M of two points A and B by entering
M = (A + B) / 2 into the Input Bar.
• You may calculate the length of a vector v using length = sqrt(v * v)
3.2.3. Lines and Axes
Lines
You can enter a line as a linear equation in x and y or in parametric form into the Input Bar.
In both cases previously defined variables (e. g. numbers, points, vectors) can be used within
the equation.
Note: You can enter a line's name at the beginning of the input followed by a colon.
Examples:
• Type in g: 3x + 4y = 2 to enter line g as a linear equation.
• Define a parameter t (e. g., t = 3) before entering line g in parametric form using
g: X = (-5, 5) + t (4, -3).
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• Define the parameters m = 2 and b = -1. Then, you can enter the equation
h: y = m*x + b to get a line h in y‐intercept‐form.
Axes
The two coordinate axes are available in commands using the names xAxis and yAxis.
Example: The command Perpendicular[A, xAxis] constructs the perpendicular line to
the x‐axis through a given point A.
3.2.4. Conic Sections
You may enter a conic section as a quadratic equation in x and y. Prior defined variables (e.
g., numbers, points, vectors) can be used within the conic's equation.
Note: The conic section's name can be entered at the beginning of the input followed by a
colon.
Examples:
• Ellipse ell: ell: 9 x^2 + 16 y^2 = 144
• Hyperbola hyp: hyp: 9 x^2 – 16 y^2 = 144
• Parabola par: par: y^2 = 4 x
• Circle c1: c1: x^2 + y^2 = 25
• Circle c2: c2: (x – 5)^2 + (y + 2)^2 = 25
Note: If you define two parameters a = 4 and b = 3 in advance, you may enter for
example an ellipse as ell: b^2 x^2 + a^2 y^2 = a^2 b^2.
3.2.5. Functions of x
To enter a function you can use previously defined variables (e. g. numbers, points, vectors)
as well as other functions.
Examples:
• Function f: f(x) = 3 x^3 – x^2
• Function g: g(x) = tan(f(x))
• Nameless function: sin(3 x) + tan(x)
Note: All available pre‐defined functions (e. g. sin, cos, tan) are described in section Pre‐
defined Functions and Operations.
In GeoGebra you can also use commands to get for example, the Integral and Derivative of a
function.
Note: You can also use the commands f'(x) or f''(x),… in order to get the derivatives of a
previously defined function f(x).
Example: Define function f as f(x) = 3 x^3 – x^2. Then, you can type in
g(x) = cos(f' (x + 2)) in order to get function g.
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Operation / Function Input
Antihyperbolic tangent atanh( )
Greatest integer less than or equal floor( )
Least integer greater than or equal ceil( )
Round round( )
3.2.7. Boolean Variables and Operations
You can use the Boolean variables true and false in GeoGebra. Just type, for example,
a = true or b = false into the Input Bar and press the Enter‐key.
Check Box and Arrow Keys
Free Boolean variables can be displayed as check boxes in the Graphics View (see tool
Check Box to Show/Hide objects). By using the arrow keys of your keyboard you may also
change Boolean variables in the Algebra View (see section Manual Animation).
Note: You may also use Boolean variables like numbers (value 0 or 1). This allows you to use
a checkbox as the dynamic speed of an animated slider allowing you to start and stop the
animation. In this case, the animation button is only shown in the Graphics View if there is
also an animated slider with static (i. e. non‐dynamic) speed.
Operations
You can use the following operations for Boolean variables and conditions in GeoGebra by
either selecting them from the list next to the Input Bar or by entering them using the
keyboard:
List Keyboard Example Object types
numbers, points,
Equal == a b or a == b
lines, conics a, b
numbers, points,
Unequal ≠ != a ≠ b or a != b
lines, conics a, b
Less than < < a < b numbers a, b
Greater than > > a > b numbers a, b
Less or equal
≤ <= a ≤ b or a <= b numbers a, b
than
Greater or
≥ >= a ≥ b or a >= b numbers a, b
equal than
And && a b or a && b Booleans a, b
Or || a b or a || b Booleans a, b
Not ¬ ! ¬a or !a Boolean a
Parallel a b lines a, b
Perpendicular a b lines a, b
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3.2.8. List Objects and Operations
Using curly braces you can create a list of several objects (e. g. points, segments, circles).
Examples:
• L = {A, B, C} gives you a list consisting of three prior defined points A, B, and C.
• L = {(0, 0), (1, 1), (2, 2)} produces a list that consists of the entered
points, as well as these nameless points.
Note: By default, the elements of this list are not shown in the Graphics View.
Compare Lists of Objects
You can compare two lists of objects by using the following syntax:
• List1 == List2: Checks if the two lists are equal and gives you true or false as a
result.
• List1 != List2: Checks if the two lists are not equal and gives you true or false as
a result.
Apply Pre‐defined Operations and Functions to Lists
Note: If you apply operations and pre‐defined functions to lists, you will always get a new
list as a result.
Addition and Subtraction examples:
• List1 + List2: Adds corresponding elements of two lists.
Note: The two lists need to be of the same length.
• List + Number: Adds the number to every element of the list.
• List1 – List2: Subtracts the elements of the second list from corresponding
elements of the first list.
Note: The lists need to be of the same length.
• List – Number: Subtracts the number from every element of the list.
Multiplication and Division examples:
• List1 * List2: Multiplies corresponding elements of two lists.
Note: The lists need to be of the same length. If the two lists are compatible
matrices, matrix multiplication is used.
• List * Number: Multiplies every list element with the number.
• List1 / List2: Divides elements of the first list by corresponding elements of the
second list.
Note: The two lists need to be of the same length.
• List / Number: Divides every list element by the number.
• Number / List: Divides the number by every element of the list.
Examples using functions:
• List^2: Squares every element of the list.
• sin(List): Applies the sine function to every element of the list.
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3.2.9. Matrix Objects and Operations
GeoGebra also supports matrices, which are represented as a list of lists that contain the
rows of the matrix.
⎧1 2 3⎫
⎪ ⎪
Example: In GeoGebra, {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} represents the matrix ⎨4 5 6⎬ .
⎪7 8 9⎪
⎩ ⎭
Matrix Operations
Addition and subtraction examples:
• Matrix1 + Matrix2: Adds the corresponding elements of two compatible
matrices.
• Matrix1 – Matrix2: Subtracts the corresponding elements of two compatible
matrices.
Multiplication examples:
• Matrix * Number: Multiplies every element of the matrix by the given number.
• Matrix1 * Matrix2: Uses matrix multiplication to calculate the resulting matrix.
Note: The rows of the first and columns of the second matrix need to have the same
number of elements.
Example: {{1, 2}, {3, 4}, {5, 6}} * {{1, 2, 3}, {4, 5, 6}} gives
you the matrix {{9, 12, 15}, {19, 26, 33}, {29, 40, 51}}.
• 2x2 Matrix * Point (or Vector): Multiplies the matrix with the given
point/vector and gives you a point as a result.
Example: {{1, 2}, {3, 4}} * (3, 4) gives you the point A = (11, 25).
• 3x3 Matrix * Point (or Vector): Multiplies the matrix with the given
point/vector and gives you a point as a result.
Example: {{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * (1, 2) gives you the
point A = (8, 20).
Note: This is a special case for affine transformations where homogenous
coordinates are used: (x, y, 1) for a point and (x, y, 0) for a vector. This example is
therefore equivalent to:
{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * {1, 2, 1}.
Other examples (see also section Matrix Commands):
• Determinant[Matrix]: Calculates the determinant for the given matrix.
• Invert[Matrix]: Inverts the given matrix
• Transpose[Matrix]: Transposes the given matrix
3.2.10. Complex Numbers and Operations
GeoGebra does not support complex numbers directly, but you may use points to simulate
operations with complex numbers.
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Example: If you enter the complex number 3 + 4i into the Input Bar, you get the point
(3, 4) in the Graphics View. This point's coordinates are shown as 3 + 4i in the Algebra View.
Note: You can display any point as a complex number in the Algebra View. Open the
Properties Dialog for the point and select Complex Number from the list of Coordinates
formats on tab Algebra.
If the variable i has not already been defined, it is recognized as the ordered pair i = (0, 1) or
the complex number 0 + 1i. This also means, that you can use this variable i in order to type
complex numbers into the Input Bar (e. g., q = 3 + 4i).
Addition and subtraction examples:
• (2 + 1i) + (1 – 2i) gives you the complex number 3 – 1i.
• (2 + 1i) - (1 – 2i) gives you the complex number 1 + 3i.
Multiplication and division examples:
• (2 + 1i) * (1 – 2i) gives you the complex number 4 – 3i.
• (2 + 1i) / (1 – 2i) gives you the complex number 0 + 1i.
Note: The usual multiplication (2, 1)*(1, -2) gives you the scalar product of the two
vectors.
Other examples:
GeoGebra also recognizes expressions involving real and complex numbers.
• 3 + (4 + 5i) gives you the complex number 7 + 5i.
• 3 - (4 + 5i) gives you the complex number ‐1 ‐ 5i.
• 3 / (0 + 1i) gives you the complex number 0 ‐ 3i.
• 3 * (1 + 2i) gives you the complex number 3 + 6i.
3.3. Commands
Using commands you can produce new and modify existing objects.
Note: A command's result may be named by entering a label followed by an equal sign (=).
In the example below, the new point is named S.
Example: To get the intersection point of two lines g and h you can enter
S = Intersect[g, h] (see command Intersect).
Note: You can also use indices within the names of objects: A1 is entered as A_1 while SAB is
created using s_{AB}.
Automatic Completion of Commands
When you type a command into GeoGebra's Input Bar, the software tries to automatically
complete the command for you. This means that after you typed in the first two letters of
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the command into the Input Bar, GeoGebra displays the first command of an alphabetically
sorted list that starts with these letters.
• In order to accept this suggestion and place the cursor in between the brackets, hit
the Enter‐key.
• If the suggested command is not the one you wanted to type in, just keep typing.
GeoGebra will adapt its suggestions to the letters you enter.
3.3.1. General Commands
ConstructionStep
ConstructionStep[]: Returns the current Construction Protocol step as a number.
ConstructionStep[Object]: Returns the Construction Protocol step for the given object
as a number.
Delete
Delete[Object]: Deletes the object and all its dependents objects.
Note: Also see tool Delete Object
Relation
Relation[Object a, Object b]: Shows a message box that gives you information
about the relation of object a and object b.
Note: This command allows you to find out whether two objects are equal, if a point
lies on a line or conic, or if a line is tangent or a passing line to a conic.
Note: Also see tool Relation
3.3.2. Boolean Commands
If
If[Condition, Object]: Yields a copy of the object if the condition evaluates to true,
and an undefined object if it evaluates to false.
If[Condition, Object a, Object b]: Yields a copy of object a if the condition
evaluates to true, and a copy of object b if it evaluates to false.
IsDefined
IsDefined[Object]: Returns true or false depending on whether the object is defined or
not.
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IsInteger
IsInteger[Number]: Returns true or false depending whether the number is an integer
or not.
3.3.3. Number Commands
AffineRatio
AffineRatio[Point A, Point B, Point C]: Returns the affine ratio λ of three
collinear points A, B, and C, where C = A + λ * AB.
Area
Area[Point A, Point B, Point C, ...]: Calculates the area of the polygon defined
by the given points A, B, C,…
Area[Conic c]: Calculates the area of a conic section c (circle or ellipse).
Note:
• In order to calculate the area between two function graphs, you need to use the
command Integral.
• Also see tool Area
AxisStep
AxisStepX[]: Returns the current step width for the x‐axis.
AxisStepY[]: Returns the current step width for the y‐axis.
Note: Together with the Corner and Sequence commands, the AxisStep commands allow you
to create custom axes (also see section Customizing Coordinate Axes and Grid).
BinomialCoefficient
BinomialCoefficient[Number n, Number r]: Calculates the binomial coefficient
n choose r.
Circumference
Circumference[Conic]: Returns the circumference of a circle or ellipse.
CrossRatio
CrossRatio[Point A, Point B, Point C, Point D]: Calculates the cross ratio λ
of four collinear points A, B, C, and D, where
λ = AffineRatio[B, C, D] / AffineRatio[A, C, D].
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Curvature
Curvature[Point, Function]: Calculates the curvature of the function in the given
point.
Curvature[Point, Curve]: Calculates the curvature of the curve in the given point.
Distance
Distance[Point A, Point B]: Yields the distance of two points A and B.
Distance[Point, Line]: Yields the distance of the point and the line.
Distance[Line g, Line h]: Yields the distance of the parallel lines g and h.
Note: The distance of intersecting lines is 0. Thus, this command is only interesting
for parallel lines.
Note: Also see tool Distance or Length
GCD
UK English: HCF
GCD[Number a, Number b]: Calculates the greatest common divisor of numbers a and b
(UK‐English: HCF = highest common factor).
GCD[List of Numbers]: Calculates the greatest common divisor of the list of numbers
(UK‐English: HCF = highest common factor).
Integer Division
Div[Number a, Number b]: Calculates the integer quotient for division of number a by
number b.
Integral
Integral[Function, Number a, Number b]: Returns the definite integral of the
function in the interval [a , b].
Note: This command also draws the area between the function graph of f and the x‐
axis.
Integral[Function f, Function g, Number a, Number b]: Yields the definite
integral of the difference f(x) ‐ g(x) in the interval [a, b].
Note: This command also draws the area between the function graphs of f and g.
Note: Also see command for Indefinite Integral
Iteration
Iteration[Function, Number x0, Number n]: Iterates the function n times using
the given start value x0.
Example: After defining f(x) = x^2 the command Iteration[f, 3, 2] gives
you the result (32)2 = 81.
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LCM
LCM[Number a, Number b]: Calculates the least common multiple of two numbers a
and b (UK English: LCM = lowest common multiple).
LCM[List of numbers]: Calculates the least common multiple of the elements of the list
(UK English: LCM = lowest common multiple).
Length
Length[Vector]: Yields the length of the vector.
Length[Point A]: Yields the length of the position vector of the given point .
Length[Function, Number x1, Number x2]: Yields the length of the function graph
in the interval [x1, x2].
Length[Function, Point A, Point B]: Yields the length of the function graph
between the two points A and B.
Note: If the given points do not lie on the function graph, their x‐coordinates are
used to determine the intervalLength[List]: Yields the length of the list which is the number of elements in the list.
Note: Also see tool Distance or Length
LinearEccentricity
LinearEccentricity[Conic]: Calculates the linear eccentricity of the conic section.
Note: The linear eccentricity is the distance between a conic's center and its focus, or
one of its two foci.
LowerSum
LowerSum[Function, Number a, Number b, Number n]: Yields the lower sum of
the given function on the interval [a, b] with n rectangles.
Note: This command draws the rectangles for the lower sum as well.
Minimum and Maximum
Min[Number a, Number b]: Yields the minimum of the given numbers a and b.
Max[Number a, Number b]: Yields the maximum of the given numbers a and b.
Modulo Function
Mod[Integer a, Integer b]: Yields the remainder when integer a is divided by integer
b.
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Parameter
Parameter[Parabola]: Returns the parameter of the parabola, which is the distance of
directrix and focus.
Perimeter
Perimeter[Polygon]: Returns the perimeter of the polygon.
Radius
Radius[Circle]: Returns the radius of the circle.
Random commands
RandomBetween[Min Integer, Max Integer]: Generates a random integer between
min and max (inclusive).
RandomBinomial[Number n of Trials, Probability p]: Generates a random
number from a binomial distribution with n trials and probability p.
RandomNormal[Mean, Standard Deviation]: Generates a random number from a
normal distribution with given mean and standard deviation.
RandomPoisson[Mean]: Generates a random number from a Poisson distribution with
given mean.
SemiMajorAxisLength
SemiMajorAxisLength[Conic]: Returns the length of the semimajor axis (half of the
major axis) of the conic section.
SemiMinorAxisLength
SemiMinorAxisLength[Conic]: Returns the length of the semiminor axis (half of the
minor axis) of the conic section.
Slope
Slope[Line]: Returns the slope of the given line.
Note: This command also draws the slope triangle whose size may be changed on tab
Style of the Properties Dialog.
Note: Also see tool Slope
TrapezoidalSum
UK English: TrapeziumSum
TrapezoidalSum[Function, Number a, Number b, Number n]: Calculates the
trapezoidal sum of the function in the interval [a, b] using n trapezoids.
Note: This command draws the trapezoids of the trapezoidal sum as well.
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UpperSum
UpperSum[Function, Number a, Number b, Number n]: Calculates the upper sum
of the function on the interval [a, b] using n rectangles.
Note: This command draws the rectangles of the upper sum as well.
3.3.4. Angle Command
Angle
Angle[Vector v1, Vector v2]: Returns the angle between two vectors v1 and v2
(between 0 and 360°) .
Angle[Line g, Line h]: Returns the angle between the direction vectors of two lines g
and h (between 0 and 360°) .
Angle[Point A, Point B, Point C]: Returns the angle enclosed by BA and BC
(between 0 and 360°), where point B is the apex.
Angle[Point A, Point B, Angle α]: Returns the angle of size α drawn from point A
with apex B.
Note: The point Rotate[A, α, B] is created as well.
Angle[Conic]: Returns the angle of twist of a conic section's major axis (see command
Axes) .
Angle[Vector]: Returns the angle between the x‐axis and given vector.
Angle[Point]: Returns the angle between the x‐axis and the position vector of the given
point.
Angle[Number]: Converts the number into an angle (result between 0 and 2pi).
Angle[Polygon]: Creates all angles of a polygon in mathematically positive orientation (i.
e., counter clockwise).
Note: If the polygon was created in counter clockwise orientation, you get the
interior angles. If the polygon was created in clockwise orientation, you get the
exterior angles.
Note: Also see tools Angle and Angle with Given Size
3.3.5. Point Commands
Center
UK English: Centre
Center[Conic]: Returns the center of a circle, ellipse, or hyperbola.
Note: Also see tool Midpoint or Center
Centroid
Centroid[Polygon]: Returns the centroid of the polygon.
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Corner
Corner[Number n of Corner]: Creates a point at the corner of the Graphics View
(n = 1, 2, 3, 4) which is never visible on screen.
Corner[Image, Number n of Corner]: Creates a point at the corner of the image
(n = 1, 2, 3, 4).
Corner[Text, Number n of Corner]: Creates a point at the corner of the text
(n = 1, 2, 3, 4).
Note: The numbering of the corners is counter‐clockwise and starts at the lower left corner.
Extremum
UK English: TurningPoint
Extremum[Polynomial]: Yields all local extrema of the polynomial function as points on
the function graph.
Focus
Focus[Conic]: Yields (all) foci of the conic section.
InflectionPoint
InflectionPoint[Polynomial]: Yields all inflection points of the polynomial as points
on the function graph.
Intersect
Intersect[Line g, Line h]: Yields the intersection point of lines g and h.
Intersect[Line, Conic]: Yields all intersection points of the line and conic section
(max. 2).
th
Intersect[Line, Conic, Number n]: Yields the n intersection point of the line and
the conic section.
Intersect[Conic c1, Conic c2]: Yields all intersection points of conic sections c1
and c2 (max. 4).
th
Intersect[Conic c1, Conic c2, Number n]: Yields the n intersection point of
conic sections c1 and c2.
Intersect[Polynomial f1, Polynomial f2]: Yields all intersection points of
polynomials f1 and f2.
th
Intersect[Polynomial f1, Polynomial f2, Number n]: Yields the n
intersection point of polynomials f1 and f2.
Intersect[Polynomial, Line]: Yields all intersection points of the polynomial and the
line.
th
Intersect[Polynomial, Line, Number n]: Yields the n intersection point of the
polynomial and the line.
Intersect[Function f, Function g, Point A]: Calculates the intersection point
of functions f and g by using Newton's method with initial point A.
Intersect[Function, Line, Point A]: Calculates the intersection point of the
function and the line by using Newton's method with initial point A.
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Note: Also see tool Intersect two Objects
Midpoint
Midpoint[Point A, Point B]: Returns the midpoint of points A and B.
Midpoint[Segment]: Returns the midpoint of the segment.
Note: Also see tool Midpoint or Center
Point
Point[Line]: Returns a point on the line.
Point[Conic]: Returns a point on the conic section.
Point[Function]: Returns a point on the function.
Point[Polygon]: Returns a point on the polygon.
Point[Vector ]: Returns a point on the vector.
Point[Point, Vector]: Creates a new point by adding the vector to the given point.
Note: Also see tool New Point
Root
Root[Polynomial]: Yields all roots of the polynomial as intersection points of the
function graph and the x‐axis.
Root[Function, Number a]: Yields one root of the function using the initial value a for
Newton's method.
Root[Function, Number a, Number b]: Yields one root of the function in the interval
[a, b] (regula falsi).
Vertex
Vertex[Conic]: Returns (all) vertices of the conic section.
3.3.6. Vector Commands
CurvatureVector
CurvatureVector[Point, Function]: Yields the curvature vector of the function in
the given point.
CurvatureVector[Point, Curve]: Yields the curvature vector of the curve in the given
point.
Direction
Direction[Line]: Yields the direction vector of the line.
Note: A line with equation ax + by = c has the direction vector (b, ‐ a).
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PerpendicularVector
PerpendicularVector[Line]: Returns the perpendicular vector of the line.
Note: A line with equation ax + by = c has the perpendicular vector (a, b).
PerpendicularVector[Vector v]: Returns the perpendicular vector of the given
vector.
Note: A vector with coordinates (a, b) has the perpendicular vector (‐b, a).
UnitPerpendicularVector
UnitPerpendicularVector[Line]: Returns the perpendicular vector with length 1 of
the given line.
UnitPerpendicularVector[Vector]: Returns the perpendicular vector with length 1
of the given vector.
UnitVector
UnitVector[Line]: Yields the direction vector with length 1 of the given line.
UnitVector[Vector]: Yields a vector with length 1, which has the same direction and
orientation as the given vector.
Vector
Vector[Point A, Point B]: Creates a vector from point A to point B.
Vector[Point]: Returns the position vector of the given point.
Note: Also see tool Vector between Two Points
3.3.7. Segment Command
Segment
Segment[Point A, Point B]: Creates a segment between two points A and B.
Segment[Point A, Number a]: Creates a segment with length a and starting point A.
Note: The endpoint of the segment is created as well.
Note: Also see tools Segment between Two Points and Segment with Given Length
from Point
3.3.8. Ray Command
Ray
Ray[Point A, Point B]: Creates a ray starting at point A through point B.
Ray[Point, Vector v]: Creates a ray starting at the given point which has the direction
vector v.
Note: Also see tool Ray through Two Points
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3.3.9. Polygon Command
Polygon
Polygon[Point A, Point B, Point C,...]: Returns a polygon defined by the given
points A, B, C,…
Polygon[Point A, Point B, Number n]: Creates a regular polygon with n vertices
(including points A and B).
Note: Also see tools Polygon and Regular Polygon
3.3.10. Line Commands
AngleBisector
AngleBisector[Point A, Point B, Point C]: Returns the angle bisector of the
angle defined by points A, B, and C.
Note: Point B is apex of this angle.
AngleBisector[Line g, Line h]: Returns both angle bisectors of the lines.
Note: Also see tool Angle Bisector
Asymptote
Asymptote[Hyperbola]: Yields both asymptotes of the hyperbola.
Axes
Axes[Conic]: Returns the major and minor axis of a conic section.
ConjugateDiameter
ConjugateDiameter[Line, Conic]: Returns the conjugate diameter of the diameter
that is parallel to the line (relative to the conic section).
ConjugateDiameter[Vector, Conic]: Returns the conjugate diameter of the diameter
that is parallel to the vector (relative to the conic section).
Directrix
Directrix[Parabola]: Yields the directrix of the parabola.
Line
Line[Point A, Point B]: Creates a line through two points A and B.
Line[Point, Parallel Line]: Creates a line through the given point parallel to the
given line.
Line[Point, Direction Vector v]: Creates a line through the given point with
direction vector v.
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Note: Also see tool Line through Two Points
MajorAxis
MajorAxis[Conic]: Returns the major axis of the conic section.
MinorAxis
MinorAxis[Conic]: Returns the minor axis of the conic section.
PerpendicularLine
PerpendicularLine[Point, Line]: Creates a line through the point perpendicular to
the given line.
PerpendicularLine[Point, Vector]: Creates a line through the point perpendicular
to the given vector.
Note: Also see tool Perpendicular Line
PerpendicularBisector
PerpendicularBisector[Point A, Point B]: Yields the perpendicular bisector of
the line segment AB.
PerpendicularBisector[Segment]: Yields the perpendicular bisector of the segment.
Note: Also see tool Perpendicular Bisector
Polar
Polar[Point, Conic]: Creates the polar line of the given point relative to the conic
section.
Note: Also see tool Polar or Diameter Line
Tangent
Tangent[Point, Conic]: Creates (all) tangents through the point to the conic section.
Tangent[Line, Conic]: Creates (all) tangents to the conic section that are parallel to
the given line.
Tangent[Number a, Function]: Creates the tangent to the function at x = a.
Tangent[Point A, Function]: Creates the tangent to the function at x = x(A).
ote: x(A) is the x‐coordinate of point A.
N
Tangent[Point, Curve]: Creates the tangent to the curve in the given point.
Note: Also see tool Tangents
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3.3.11. Conic Section Commands
Circle
Circle[Point M, Number r]: Yields a circle with midpoint M and radius r.
Circle[Point M, Segment]: Yields a circle with midpoint M whose radius is equal to
the length of the given segment.
Circle[Point M, Point A]: Yields a circle with midpoint M through point A.
Circle[Point A, Point B, Point C]: Yields a circle through the given points A, B
and C.
Note: Also see tools Compass, Circle with Center through Point, Circle with Center
and Radius, and Circle through Three Points
Conic
Conic[Point A, Point B, Point C, Point D, Point E]: Returns a conic section
through the five given points A, B, C, D, and E.
Note: If four of the points lie on one line the conic section is not defined.
Note: Also see tool Conic through Five Points
Ellipse
Ellipse[Point F, Point G, Number a]: Creates an ellipse with focal points F and G
and semimajor axis length a.
Note: Condition: 2a > Distance[F, G]
Ellipse[Point F, Point G, Segment]: Creates an ellipse with focal points F and G
where the length of the semimajor axis equals the length of the given segment.
Ellipse[Point F, Point G, Point A]: Creates an ellipse with foci F and G passing
through point A.
Note: Also see tool Ellipse
Hyperbola
Hyperbola[Point F, Point G, Number a]: Creates a hyperbola with focal points F
and G and semimajor axis length a.
Note: Condition: 0 < 2a < Distance[F, G]
Hyperbola[Point F, Point G, Segment]: Creates a hyperbola with focal points F.
and G where the length of the semimajor axis equals the length of segment s.
Hyperbola[Point F, Point G, Point A]: Creates a hyperbola with foci F and G
passing through point A.
Note: Also see tool Hyperbola
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OsculatingCircle
OsculatingCircle[Point, Function]: Yields the osculating circle of the function in
the given point.
OsculatingCircle[Point, Curve]: Yields the osculating circle of the curve in the
given point.
Parabola
Parabola[Point F, Line g]: Returns a parabola with focal point F and directrix g.
Note: Also see tool Parabola
3.3.12. Function Commands
Conditional Functions
You can use the Boolean command If in order to create a conditional function.
Note: You can use derivatives and integrals of such functions and intersect conditional
functions like "normal" functions.
Examples:
• f(x) = If[x < 3, sin(x), x^2] gives you a function that equals sin(x) for
x < 3 and x2 for x ≥ 3.
• a 3 ˄ b ≥ 0 tests whether "a equals 3 and b is greater than or equal to 0".
Note: Symbols for conditional statements (e. g., , ˄, ≥) can be found in the list next to the
right of the Input Bar.
Derivative
Derivative[Function]: Returns the derivative of the function.
Derivative[Function, Number n]: Returns the nth derivative of the function.
Note: You can use f'(x) instead of Derivative[f]as well as f''(x) instead of
Derivative[f, 2] and so on.
Expand
Expand[Function]: Multiplies out the brackets of the expression.
Example: Expand[(x + 3)(x - 4)] gives you f(x) = x2 ‐ x ‐ 12
Factor
UK English: Factorise
Factor[Polynomial]: Factors the polynomial.
Example: Factor[x^2 + x - 6] gives you f(x) = (x‐2)(x+3)
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Function
Function[Function, Number a, Number b]: Yields a function graph, that is equal to
f on the interval [a, b] and not defined outside of [a, b].
Note: This command should be used only in order to display functions in a certain interval.
Example: f(x) = Function[x^2, -1, 1] gives you the graph of function x2 in the
interval [‐1, 1]. If you then type in g(x) = 2 f(x) you will get the function g(x) = 2 x2, but
this function is not restricted to the interval [‐1, 1].
Integral
Integral[Function]: Yields the indefinite integral for the given function.
Note: Also see command for Definite integral
Polynomial
Polynomial[Function]: Yields the expanded polynomial function.
Example: Polynomial[(x - 3)^2] yields x2 ‐ 6x + 9.
Polynomial[List of n points]: Creates the interpolation polynomial of degree n‐1
through the given n points.
Simplify
Simplify[Function]: Simplifies the terms of the given function if possible.
Examples:
• Simplify[x + x + x] gives you a function f(x) = 3x.
• Simplify[sin(x) / cos(x)] gives you a function f(x) = tan(x).
• Simplify[-2 sin(x) cos(x)] gives you a function f(x) = sin(‐2 x).
TaylorPolynomial
TaylorPolynomial[Function, Number a, Number n]: Creates the power series
expansion for the given function about the point x = a to order n.
3.3.13. Parametric Curve Command
Curve
Curve[Expression e1, Expression e2, Parameter t, Number a, Number
b]: Yields the Cartesian parametric curve for the given x‐expression e1 and y‐
expression e2 (using parameter t) within the given interval [a, b].
Example: Input of c = Curve[2 cos(t), 2 sin(t), t, 0, 2 pi] creates a
circle with radius 2 around the origin of the coordinate system.
Note: Parametric curves can be used with pre‐defined functions and arithmetic operations.
Example: Input c(3) returns the point at parameter position 3 on curve c.
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Note: Using the mouse you can also place a point on a curve using tool New Point or
command Point. Since the parameters a and b are dynamic you could use slider variables as
well (see tool Slider).
Commands for Parametric Curves
Curvature[Point, Curve]: Calculates the curvature of the curve in the given point.
CurvatureVector[Point, Curve]: Yields the curvature vector of the curve in the given
point.
Derivative[Curve]: Returns the derivative of the parametric curve.
th
Derivative[Curve, Number n]: Returns the n derivative of the parametric curveOsculatingCircle[Point, Curve]: Yields the osculating circle of the curve in the
given point.
Tangent[Point, Curve]: Creates the tangent to the curve in the given point.
3.3.14. Arc and Sector Commands
Note: The algebraic value of an arc is its length and the value of a sector is its area.
Arc
Arc[Conic, Point A, Point B]: Returns a conic section arc between two points A
and B on the conic section c.
Note: This only works for a circle or ellipse.
Arc[Conic, Number t1, Number t2]: Returns a conic section arcCircularArc
CircularArc[Point M, Point A, Point B]: Creates a circular arc with midpoint M
between points A and B.
Note: Point B does not have to lie on the arc.
Note: Also see tool Circular Arc with Center between Two Points
CircularSector
CircularSector[Point M, Point A, Point B]: Creates a circular sector with
midpoint M between two points A and B.
Note: Point B does not have to lie on the arc of the sector.
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Note: Also see tool Circular Sector with Center between Two Points
CircumcircularArc
CircumcircularArc[Point A, Point B, Point C]: Creates a circular arc through
three points A, B, and C, where A is the starting point and C is the endpoint of the
circumcircular arc.
Note: Also see tool Circumcircular Arc through Three Points
CircumcircularSector
CircumcircularSector[Point A, Point B, Point C]: Creates a circular sector
whose arc runs through the three points A, B, and C. Point A is the starting point and
point C is the endpoint of the arc.
Note: Also see tool Circumcircular Sector through Three Points
Sector
Sector[Conic, Point A, Point B]: Yields a conic section sector between two points
A and B on the conic section.
Note: This works only for a circle or ellipse.
Sector[Conic, Number t1, Number t2]: Yields a conic section sectorSemicircle
Semicircle[Point A, Point B]: Creates a semicircle above the segment AB.
Note: Also see tool Semicircle
3.3.15. Text Commands
FormulaText
FormulaText[Object]: Returns the formula for the object as a LaTeX text.
Example: If a = 2 and f(x) = a x2, then FormulaText[f] returns 2 x2 (as a LaTeX
text).
FormulaText[Object, Boolean]: Returns the formula for the object as LaTeX text. The
Boolean variable determines if values are substituted for variables (true) or if
variable names are shown in the text (false).
2
Examples: If a = 2 and f(x) = a x , then
55
2
FormulaText[f, true] returns 2 x (as a LaTeX text).
FormulaText[f, false] returns a x2 (as a LaTeX text).
FractionText
FractionText[Number]: Converts the number to a fraction, which is displayed as a
(LaTeX) text object in the Graphics View.
Example: If a: y = 1.5 x + 2 is a line, then FractionText[Slope[a]] gives you the
fraction 3/2 as a text.
LetterToUnicode
LetterToUnicode["Letter"]: Turns a single letter into its Unicode number
Note: The letter needs to be in between a set of quotation marks.
Example: LetterToUnicode["a"] returns the number 97
Name
Name[Object]: Returns the name of an object as a text in the Graphics View.
Note: Use this command in dynamic text for objects that might be renamed. The
Name command is the opposite of the Object command.
Object
Object[Name of Object as Text]: Returns the object for a given name. The result is
always a dependent object.
Note: The Object command is the opposite of the Name command.
Example: If points A1, A2, ... , A20 exist and slider n = 2, then Object["A" + n]
gives you a copy of point A2.
TableText
TableText[List 1, List 2, List 3,...]: Creates a text that contains a table of
the list objects.
Note: By default, every list is displayed in a new column of the table.
Examples:
• TableText[{x^2, 4}, {x^3, 8}, {x^4, 16}]
creates a table as a text object with three rows and two columns. All items of
the table are left aligned.
• TableText[Sequence[i^2, i, 1, 10]]
creates a table as a text object with one row. All items of the table are left
aligned.
TableText[List 1, List 2, List 3,..., "Alignment of text"]: Creates a
text that contains a table of the list objects. The optional text "Alignment of text"
controls the orientation and alignment of the table text.
Note: Possible values are "vl", "vc", "vr", "v", "h", "hl", "hc", "hr". Default is "hl".
• "v" = vertical, i. e. lists are columns
• "h" = horizontal, i. e. lists are rows
• "l" = left aligned
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• "r" = right aligned
• "c" = centered
Examples:
• TableText[{1,2,3,4},{1,4,9,16},"v"]
creates a text with two columns and four rows whose elements are left
aligned.
• TableText[{1,2,3,4},{1,4,9,16},"h"]
creates a text with two rows and four columns whose elements are left
aligned.
• TableText[{11.2,123.1,32423.9,"234.0"},"vr"]
creates a text with one column whose elements are right aligned.
Text
Text[Object]: Returns the formula for the object as a text object.
Note: By default, values are substituted for variables.
Example: If a = 2 and c = a2, then Text[c] returns the text "4".
Text[Object, Boolean]: Returns the formula for the object as a text object. The
Boolean variable determines if values are substituted for variables (true) or if
variable names are shown in the text (false).
2
Example: If a = 2 and c = a , then
Text[c, true] returns the text "4".
Text[c, false] returns the text "a2".
Text[Object, Point]: Returns the formula for the object as a text object at the position
of the given point.
Example: Text["hello", (2, 3)] draws the text at the position (2, 3).
Text[Object, Point, Boolean]: Returns the formula for the object as a text object at
the position of the given point. The Boolean variable determines if values are
substituted for variables (true) or if variable names are shown in the text (false).
TextToUnicode
TextToUnicode["Text"]: Turns the text into a list of Unicode numbers, one for each
character.
Examples:
• TextToUnicode["Some text"] gives you the list of Unicode numbers
{83, 111, 109, 101, 32, 116, 101, 120, 116}.
• If text1 is "hello", then TextToUnicode[text1] gives you the list of
Unicode numbers {104, 101, 108, 108, 111}.
UnicodeToLetter
UnicodeToLetter[Integer]: Converts the integer Unicode number back into a letter
which is displayed as a text object in the Graphics View.
Example: UnicodeToLetter[97] gives you the text "a".
57
UnicodeToText
UnicodeToText[List of Integers]: Converts the integer Unicode numbers back into
text.
Example: UnicodeToText[{104, 101, 108, 108, 111}] gives you the text
"hello".
3.3.16. Locus Command
Locus
Locus[Point Q, Point P]: Returns the locus line of point Q which depends on point P.
Note: Point P has to be a point on an object (e. g. line, segment, circle).
Note: Also see tool Locus
3.3.17. List and Sequence Commands
Append
Append[List, Object]: Appends the object to the list.
Example: Append[{1, 2, 3}, 4] gives you {1, 2, 3, 4}.
Append[Object, List]: Appends the list to the object.
Example: Append[4, {1, 2, 3}] gives you {4, 1, 2, 3}.
CountIf
CountIf[Condition, List]: Counts the number of elements in the list satisfying the
condition.
Examples:
• CountIf[x < 3, {1, 2, 3, 4, 5}] gives you the number 2.
• CountIf[x<3, A1:A10] where A1:A10 is a range of cells in the
spreadsheet, counts all cells whose values are less than 3.
Element
th
Element[List, Number n]: Yields the n element of the list.
Note: The list can contain only elements of one object type (e. g., only numbers or
only points).
First
First[List]: Returns the first element of the list.
First[List, Number n of elements]: Returns a new list that contains just the first n
elements of the list.
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Insert
Insert[Object, List, Position]: Inserts the object in the list at the given position.
Example: Insert[x^2, {1, 2, 3, 4, 5}, 3] places x2 at the third position
and gives you the list {1, 2, x2, 3, 4, 5}.
Note: If the position is a negative number, then the position is counted from the
right.
Example: Insert[x^2, {1, 2, 3, 4, 5}, -1] places x2 at the end of the list
and gives you the list {1, 2, 3, 4, 5, x2}.
Insert[List 1, List 2, Position]: Inserts all elements of list1 in list2 at the given
position.
Example: Insert[{11, 12}, {1, 2, 3, 4, 5}, 3] places the elements of
list1 at the third (and following) position(s) of list2 and gives you the list
{1, 2, 11, 12, 3, 4, 5}.
Note: If the position is a negative number, then the position is counted from the
right.
Example: Insert[{11, 12}, {1, 2, 3, 4, 5}, -2] places the elements of
list1 at the end of list2 before its last element and gives you {1, 2, 3, 4, 11, 12, 5}.
Intersection
Intersection[List 1, List 2]: Gives you a new list containing all elements that are
part of both lists.
IterationList
IterationList[Function, Number x0, Number n]:
Gives you a list of length n+1 whose elements are iterations of the function starting
with the value x0.
Example: After defining function f(x) = x^2 the command
L = IterationList[f, 3, 2] gives you the list L = {3, 9, 81}.
Join
Join[List 1, List 2, ...]: Joins the two (or more) lists.
Note: The new list contains all elements of the initial lists even if they are the same.
The elements of the new list are not re‐ordered.
Example: Join[{5, 4, 3}, {1, 2, 3}] creates the list {5, 4, 3, 1, 2, 3}.
Join[List of lists]: Joins the sub‐lists into one longer list.
Note: The new list contains all elements of the initial lists even if they are the same.
The elements of the new list are not re‐ordered.
Examples:
• Join[{{1, 2}}] creates the list {1, 2}.
• Join[{{1, 2, 3}, {3, 4}, {8, 7}}] creates the list
{1, 2, 3, 3, 4, 8, 7}.
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KeepIf
KeepIf[Condition, List]: Creates a new list that only contains those elements of the
initial list that fulfill the condition.
Example: KeepIf[x<3, {1, 2, 3, 4, 1, 5, 6}] returns the new list {1, 2, 1}.
Last
Last[List]: Returns the last element of the list.
Last[List, Number n of Elements]: Returns a list containing just the last n elements
of the list.
Length
Length[List]: Yields the length of the list, which is the number of list elements.
Min
Min[List]: Returns the minimal element of the list.
Max
Max[List]: Returns the maximal element of the list.
Product
Product[List of Numbers]: Calculates the product of all numbers in the list.
RemoveUndefined
RemoveUndefined[List]: Removes undefined objects from a list.
Example: RemoveUndefined[Sequence[(-1)^i, i, -3, -1, 0.5]] removes
the second and fourth element of the sequence which have a non‐integer exponent
and therefore, are undefined.
Reverse
Reverse[List]: Reverses the order of a list.
Sequence
Sequence[Expression, Variable i, Number a, Number b]: Yields a list of
objects created using the given expression and the index i that ranges from number
a to number b.
Example: L = Sequence[(2, i), i, 1, 5] creates a list of points whose y‐
coordinates range from 1 to 5: L = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}.
Sequence[Expression, Variable i, Number a, Number b, Increment]: Yields
a list of objects created using the given expression and the index i that ranges from
number a to number b with given increment.
Example: L = Sequence[(2, i), i, 1, 3, 0.5] creates a list of points whose
60
y‐coordinates range from 1 to 3 with an increment of 0.5:
L = {(2, 1), (2, 1.5), (2, 2), (2, 2.5), (2, 3)}.
Note: Since the parameters a and b are dynamic you could use slider variables as well.
Sort
Sort[List]: Sorts a list of numbers, text objects, or points.
Note: Lists of points are sorted by x‐coordinates.
Examples:
• Sort[{3, 2, 1}] gives you the list {1, 2, 3}.
• Sort[{"pears", "apples", "figs"}] gives you the list elements in
alphabetical order.
• Sort[{(3, 2), (2, 5), (4, 1)}] gives you {(2, 5), (3, 2), (4, 1)}.
Sum
Sum[List]: Calculates the sum of all list elements.
Note: This command works for numbers, points, vectors, text, and functions.
Examples:
• Sum[{1, 2, 3}] gives you a number a = 6.
• Sum[{x^2, x^3}] gives you f(x) = x2 + x3.
• Sum[Sequence[i,i,1,100]] gives you a number a = 5050.
• Sum[{(1, 2), (2, 3)}] gives you a point A = (3, 5).
• Sum[{(1, 2), 3}] gives you point B = (4, 2).
• Sum[{"a","b","c"}] gives you the text "abc".
Sum[List, Number n of Elements]: Calculates the sum of the first n list elements.
Note: This command works for numbers, points, vectors, text, and functions.
Example: Sum[{1, 2, 3, 4, 5, 6}, 4] gives you the number a = 10.
Take
Take[List, Start Position m, End Position n]: Returns a list containing the
elements from positions m to n of the initial list.
Union
Union[List 1, List 2]: Joins the two lists and removes elements that appear multiple
times.
3.3.18. Geometric Transformation Commands
Dilate
UK English: Enlarge
Dilate[Point A, Number, Point S]: Dilates point A from point S using the given
factor.
61
Dilate[Line, Number, Point S]: Dilates the line from point S using the given factor.
Dilate[Conic, Number, Point S]: Dilates the conic section from point S using the
given factor.
Dilate[Polygon, Number, Point S]: Dilates the polygon from point S using the given
factor.
Note: New vertices and segments are created too.
Dilate[Image, Number, Point S]: Dilates the image from point S using the given
factor.
Note: Also see tool Dilate Object from Point
Reflect
Reflect[Point A, Point B]: Reflects point A about point B.
Reflect[Line, Point]: Reflects the line about the given point.
Reflect[Conic, Point]: Reflects the conic section about the given point.
Reflect[Polygon, Point]: Reflects the polygon about the given point.
Note: New vertices and segments are created as well.
Reflect[Image, Point]: Reflects the image about the given point.
Reflect[Point, Line]: Reflects the point about the given line.
Reflect[Line g, Line h]: Reflects line g about line h.
Reflect[Conic, Line]: Reflects the conic section about the line.
Reflect[Polygon, Line]: Reflects the polygon about the line.
Note: New vertices and segments are created as well.
Reflect[Image, Line]: Reflects the image about the line.
Reflect[Point, Circle]: Inverts the point in the circle.
Note: Also see tools Reflect Object about Point; Reflect Object about Line;
Reflect Point about Circle
Rotate
Rotate[Point, Angle]: Rotates the point by the angle around the axis origin.
Rotate[Vector, Angle]: Rotates the vector by the angle around the starting point of
the vector.
Rotate[Line, Angle]: Rotates the line by the angle around the axis origin.
Rotate[Conic, Angle]: Rotates the conic section by the angle around the axis origin.
Rotate[Polygon, Angle]: Rotates the polygon by the angle around the axis origin.
Note: New vertices and segments are created as well.
Rotate[Image, Angle]: Rotates the image by the angle around the axis origin.
Rotate[Point A, Angle, Point B]: Rotates point A by the angle around point B.
Rotate[Line, Angle, Point]: Rotates the line by the angle around the point.
Rotate[Vector, Angle, Point]: Rotates the vector by the angle around the point.
Rotate[Conic, Angle, Point]: Rotates the conic section by the angle around the
point.
Rotate[Polygon, Angle, Point]: Rotates the polygon by the angle around point B.
Note: New vertices and segments are created as well.
Rotate[Image, Angle, Point]: Rotates the image by the angle around the point.
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Note: Also see tool Rotate Object around Point by Angle
Translate
Translate[Point, Vector ]: Translates the point by the vector.
Translate[Line, Vector]: Translates the line by the vector.
Translate[Conic, Vector]: Translates the conic by the vector.
Translate[Function, Vector]: Translates the function by the vector.
Translate[Polygon, Vector]: Translates the polygon by the vector.
Note: New vertices and segments are created as well.
Translate[Image, Vector]: Translates the image by the vector.
Translate[Vector, Point]: Translates the vector v to point.
Note: Also see tool Translate Object by Vector
3.3.19. Statistics Commands
BarChart
BarChart[Start Value, End Value, List of Heights]: Creates a bar chart over
the given interval where the number of bars is determined by the length of the list
whose elements are the heights of the bars.
Example: BarChart[10, 20, {1,2,3,4,5} ] gives you a bar chart with five
bars of specified height in the interval [10, 20].
BarChart[Start Value a, End Value b, Expression, Variable k, From
Number c, To Number d]: Creates a bar chart over the given interval [a, b], that
calculates the bars' heights using the expression whose variable k runs from number
c to number d.
Example: If p = 0.1, q = 0.9, and n = 10 are numbers, then
BarChart[ -0.5, n + 0.5,
BinomialCoefficient[n,k]*p^k*q^(n-k), k, 0, n ] gives you a bar chart
in the interval [‐0.5, n+0.5]. The heights of the bars depend on the probabilities
calculated using the given expression.
BarChart[Start Value a, End Value b, Expression, Variable k, From
Number c, To Number d, Step Width s]: Creates a bar chart over the given
interval [a, b], that calculates the bars' heights using the expression whose variable k
runs from number c to number d using step width s.
BarChart[List of Raw Data, Width of Bars]: Creates a bar chart using the given
raw data whose bars have the given width.
Example: BarChart[ {1,1,1,2,2,2,2,2,3,3,3,5,5,5,5}, 1]
BarChart[List of Data, List of Frequencies]: Creates a bar chart using the list
of data with corresponding frequencies.
Note: The List of data must be a list where the numbers go up by a constant amount.
Examples:
• BarChart[{10,11,12,13,14}, {5,8,12,0,1}]
• BarChart[{5, 6, 7, 8, 9}, {1, 0, 12, 43, 3}]
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• BarChart[{0.3, 0.4, 0.5, 0.6}, {12, 33, 13, 4}]
BarChart[List of Data , List of Frequencies, Width of Bars w]: Creates
a bar chart using the list of data and corresponding frequencies whose bars are of
width w.
Note: The List of data must be a list where the numbers go up by a constant amount
Examples:
• BarChart[{10,11,12,13,14}, {5,8,12,0,1}, 0.5] leaves gaps
between bars.
• BarChart[{10,11,12,13,14}, {5,8,12,0,1}, 0] produces a line
graph.
BoxPlot
BoxPlot[yOffset, yScale, List of Raw Data]: Creates a box plot using the given
raw data and whose vertical position in the coordinate system is controlled by
variable yOffset and whose height is influenced by factor yScale.
Example: BoxPlot[0, 1, {2,2,3,4,5,5,6,7,7,8,8,8,9}]
BoxPlot[yOffset, yScale, Start Value a, Q1, Median, Q3, End Value
b]: Creates a box plot for the given statistical data in interval [a, b].
CorrelationCoefficient
CorrelationCoefficient[List of x-Coordinates, List of y-
Coordinates]: Calculates the product moment correlation coefficient using the
given x‐ and y‐coordinates.
CorrelationCoefficient[List of Points]: Calculates the product moment
correlation coefficient using the coordinates of the given points.
Covariance
Covariance[List 1 of Numbers, List 2 of Numbers]: Calculates the covariance
using the elements of both lists.
Covariance[List of Points]: Calculates the covariance using the x‐ and y‐coordinates
of the points.
FitLine
FitLine[List of Points]: Calculates the y on x regression line of the points.
FitLineX[List of Points]: Calculates the x on y regression line of the points.
Note: Also see tool Best Fit Line
Other Fit Commands
FitExp[List of Points]: Calculates the exponential regression curve.
FitLog[List of Points]: Calculates the logarithmic regression curve.
FitLogistic[List of Points]: Calculates the regression curve in the form
a/(1+b ^( ‐kx)).
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Note: The first and last data point should be fairly close to the curve. The list should
have at least 3 points, preferably more.
FitPoly[List of Points, Degree n of Polynomial]: Calculates the regression
polynomial of degree n.
b
FitPow[List of Points]: Calculates the regression curve in the form a x .
Note: All points used need to be in the first quadrant of the coordinate system.
FitSin[List of Points]: Calculates the regression curve in the form
a + b sin(cx + d).
Note: The list should have at least 4 points, preferably more. The list should cover at
least two extremal points. The first two local extremal points should not be too
different from the absolute extremal points of the curve.
Histogram
Histogram[List of Class Boundaries, List of Heights]: Creates a histogram
with bars of the given heights. The class boundaries determine the width and
position of each bar of the histogram.
Example: Histogram[{0, 1, 2, 3, 4, 5}, {2, 6, 8, 3, 1}] creates a
histogram with 5 bars of the given heights. The first bar is positioned at the interval
[0, 1], the second bar is positioned at the interval
[1, 2], and so on.
Histogram[List of Class Boundaries, List of Raw Data]: Creates a
histogram using the raw data. The class boundaries determine the width and
position of each bar of the histogram and are used to determine how many data
elements lie in each class.
Example: Histogram[{1, 2, 3, 4},{1.0, 1.1, 1.1, 1.2, 1.7, 2.2,
2.5, 4.0}] creates a histogram with 3 bars, with the heights 5 (first bar), 2
(second bar), and 1 (third bar).
InverseNormal
InverseNormal[Mean μ, Standard Deviation σ, Probability P]: Calculates
the function Φ‐1(P) * σ + μ where Φ ‐1 is the inverse of the probability density
function Φ for N(0,1).
Note: Returns the x‐coordinate with the given probability to the left under the
normal distribution curve.
Mean commands
Mean[List of Numbers]: Calculates the mean of the list elements.
MeanX[List of Points]: Calculates the mean of the x‐coordinates of the points in the
list.
MeanY[List of Points]: Calculates the mean of the y‐coordinates of the points in the
list.
Median
Median[List of Numbers]: Determines the median of the list elements.
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Mode
Mode[List of Numbers]: Determines the mode(s) of the list elements.
Examples:
• Mode[{1,2,3,4}] returns an empty list {}.
• Mode[{1,1,1,2,3,4}] returns the list {1} .
• Mode[{1,1,2,2,3,3,4}] returns the list {1, 2, 3}.
Normal
Normal[Mean μ, Standard Deviation σ, Variable Value x]: Calculates the
function Φ((x – μ) / σ) where Φ is the probability density function for N(0,1) .
Note: Returns the probability for a given x‐coordinate value (or area under the
normal distribution curve to the left of the given x‐coordinate).
Quartile commands
Q1[List of Numbers]: Determines the lower quartile of the list elements.
Q3[List of Numbers]: Determines the upper quartile of the list elements.
SD
SD[List of Numbers]: Calculates the standard deviation of the numbers in the list.
Sigma commands
SigmaXX[List of Numbers]: Calculates the sum of squares of the given numbers.
Example: In order to work out the variance of a list you may use
SigmaXX[list]/Length[list] - Mean[list]^2.
SigmaXX[List of Points]: Calculates the sum of squares of the x‐coordinates of the
given points.
SigmaXY[List of x-Coordinates, List of y-Coordinates]: Calculates the sum
of the products of the x‐ and y‐coordinates.
SigmaXY[List of Points]: Calculates the sum of the products of the x‐ and y‐
coordinates.
Example: You can work out the covariance of a list of points using
SigmaXY[list]/Length[list] - MeanX[list] * MeanY[list].
SigmaYY[List of Points]: Calculates the sum of squares of y‐coordinates of the given
points.
Commands for statistic quantities
2
Sxx[List of Numbers]: Calculates the statistic Σ(x ) ‐ Σ(x) * Σ(x)/n.
Sxx[List of Points]: Calculates the statistic Σ(x2) ‐ Σ(x) * Σ(x)/n using the x‐coordinates
of the given points.
Sxy[List of Numbers, List of Numbers]: Calculates the statistic
Σ(xy) ‐ Σ(x) * Σ(y)/n.
Sxy[List of Points]: Calculates the statistic Σ(xy) ‐ Σ(x) * Σ(y)/n.
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2
Syy[List of Points]: Calculates the statistic Σ(y ) ‐ Σ(y) * Σ(y)/n using the y‐coordinates
of the given points.
Note: These quantities are simply unnormalized forms of the variances and covariance of X
and Y given by Sxx = N var(X), Syy = N var(Y), and Sxy = N cov(X, Y).
Example: You can work out the correlation coefficient for a list of points using
Sxy[list] / sqrt(Sxx[list] Syy[list]).
Variance
Variance[List of Numbers]: Calculates the variance of list elements.
3.3.20. Spreadsheet Commands
CellRange
CellRange[Start Cell, End Cell]: Creates a list containing the cell values in this cell
range.
Example: Enter the following values into the corresponding spreadsheet cells:
A1 = 1, A2 = 4, A3 = 9. The command CellRange[A1, A3] then gives you the
list {1, 4, 9}.
Column
Column[Spreadsheet Cell]: Returns the column of the cell as a number (starting at 1).
Example: Column[B3] gives you number a = 2 since column B is the second column
of the spreadsheet.
ColumnName
ColumnName[Spreadsheet Cell]: Returns the column name of the cell as a text.
Example: ColumnName[A1] gives you a text "A" in the Graphics View.
Row
Row[Spreadsheet Cell]: Returns the row number of a spreadsheet cell (starting at 1).
Example: Row[B3] gives you number a = 3.
3.3.21. Matrix Commands
Determinant
Determinant[Matrix]: Returns the determinant of the matrix.
Example: Determinant[{{1, 2}, {3, 4}}] gives you the number a = ‐2.
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4. Menu Items
4.1. File Menu
New Window
Keyboard shortcut: Ctrl‐N (Mac OS: Cmd‐N)
This menu item opens a new GeoGebra window that uses the default settings of the
GeoGebra user interface.
Note: If you change and save some of these settings, the new GeoGebra window will open
using your customized settings.
New
This menu item opens a new and empty user interface in the same GeoGebra window. You
are asked if you would like to save the existing construction before opening the new user
interface.
Note: The new user interface adopts the settings used for the prior construction. For
example, if the coordinate axes were hidden before selecting the menu item New, the axes
will be hidden in the new user interface as well.
Open…
Keyboard shortcut: Ctrl‐O (Mac OS: Cmd‐O)
This menu item allows you to open a GeoGebra file (file name extension GGB) that is saved
on your computer.
Note: In order to open a GeoGebra file you can also drag it with the mouse to the GeoGebra
window and drop it there.
Save
Keyboard shortcut: Ctrl‐S (Mac OS: Cmd‐S)
This menu item allows you to save your current construction as a GeoGebra file (file name
extension GGB) on your computer.
Note: If the file was saved before, this menu item overwrites the old file by using the same
file name.
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Save As…
This menu item allows you to save your current construction as a GeoGebra file (file name
extension GGB). You will be asked to enter a new name for your GeoGebra file before it is
saved on your computer.
Print Preview
Keyboard shortcut: Ctrl‐P (Mac OS: Cmd‐P)
This menu item opens the Print Preview window for the Graphics View. You may specify
Title, Author, Date and the Scale of your printout (in cm).
Note: Press the Enter‐key after you made a change in order to update the preview of your
printout.
Export ‐ Dynamic Worksheet as Webpage (html)…
Keyboard shortcut: Ctrl‐Shift‐W (Mac OS: Cmd‐Shift‐W)
This menu item allows you to export your current construction as a webpage (HTML) in
order to create a so called Dynamic Worksheet.
Please see section Creating Interactive Webpagesfor more information.
Export ‐ Graphics View as Picture (png, eps)…
Keyboard shortcut: Ctrl‐Shift‐P (Mac OS: Cmd‐Shift‐P)
This menu item allows you to save GeoGebra's Graphics View as a picture file on your
computer. In the appearing dialog window, you can select the picture file Format, and
change the Scale (in cm) and Resolution (in dpi) of the picture.
When exporting the Graphics View as a picture you can choose out of the following formats:
• PNG – Portable Network Graphics: This is a pixel graphics format. The higher the
resolution (dpi), the better the quality (300dpi will usually suffice). PNG graphics
should not be scaled subsequently to avoid a loss of quality.
PNG graphic files are well suited for the use on web pages (HTML) and in word
processing documents.
Note: Whenever you insert a PNG graphic file into a word processing document
(menu Insert, Image from file) make sure that the size is set to 100 %. Otherwise the
given scale (in cm) would be changed.
• EPS – Encapsulated Postscript: This is a vector graphics format. EPS pictures may be
scaled without loss of quality. EPS graphic files are well suited for the use with vector
graphics programs (e. g., Corel Draw) and professional text processing systems (e. g.,
LaTeX).
The resolution of an EPS graphic is always 72dpi. This value is only used to calculate
the true size of an image in centimeters and has no effect on the image's quality.
Note: The transparency effect with filled polygons or conic sections is not possible
with EPS.
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• PDF – Portable Document Format (see EPS format above)
Note: In SVG and PDF export you have the option to export text as editable text or
shapes. This stores the text either as text (this lets you edit the text in e. g., InkScape)
or as Bézier curves (this guarantees that the text looks the same even if the correct
font is not installed).
• SVG – Scaleable Vector Graphic (see EPS format above)
• EMF – Enhanced Meta Format (see EPS format above)
Export ‐ GraphicsExport ‐ Graphics View as PSTricks…
Keyboard shortcut: Ctrl‐Shift‐T (Mac OS: Cmd‐Shift‐T)
This menu item allows you to save the Graphics View as a PSTricks picture file, which is a
LaTeX picture format.
Export ‐ Graphics View as PGF/TikZ……
This menu item allows you to save the Graphics View as a PGF/TikZ picture file, which is a
LaTeX picture format.
Close
Keyboard shortcut: Alt‐F4 (Mac OS: Cmd‐W)
This menu item closes the GeoGebra window. If you didn't save your construction prior to
selecting Close, you are asked if you would like to do so.
4.2. Edit Menu
Undo
Keyboard shortcut: Ctrl‐Z (Mac OS: Cmd‐Z)
This menu item allows you to undo your activities step by step.
Note: You can also use the Undo button to the right of the Toolbar.
Redo
Keyboard shortcut: Ctrl‐Y (Mac OS: Cmd‐Shift‐Z)
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This menu item allows you to redo your activities step by step.
Note: You can also use the Redo button to the right of the Toolbar.
Delete
Keyboard shortcut: Delete key
This menu items allows you to delete selected objects and their dependent objects.
Note: You need to select the objects you want to delete first (e. g., use a selection
rectangle).
Select All
Keyboard shortcut: Ctrl‐A (Mac OS: Cmd‐A)
This menu item allows you to select all objects used in your construction.
Select Current Layer
Keyboard shortcut: Ctrl‐L (Mac OS: Cmd‐L)
This menu item allows you to select all objects that are on the same layer as a selected
object.
Note: You need to select one object that lies on the desired layer prior to using this menu
item.
Select Descendants
Keyboard shortcut: Ctrl‐Shift‐Q (Mac OS: Cmd‐Shift‐Q)
This menu item allows you to select all objects that depend on the selected object.
Note: You need to select the parent object prior to using this menu item.
Select Ancestors
Keyboard shortcut: Ctrl‐Q
This menu item allows you to select all objects that are ancestors of the selected object,
meaning all objects the selected one depends on.
Note: You need to select the dependent object prior to using this menu item.
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Properties…
Keyboard shortcut: Ctrl‐E (Mac OS: Cmd‐E)
This menu item opens the Properties Dialog which allows you to modify the properties of all
objects used in the GeoGebra file.
4.3. View Menu
Axes
This menu item allows you to show or hide the coordinate axes in the Graphics View.
Note: You can use the Properties Dialog for Graphics View in order to customize the
coordinate axes.
Grid
This menu item allows you to show or hide the coordinate grid in the Graphics View.
Note: You can use the Properties Dialog for Graphics View in order to customize the
coordinate grid.
Algebra View
Keyboard shortcut: Ctrl‐Shift‐A (Mac OS: Cmd‐Shift‐A)
This menu item allows you to show or hide the Algebra View.
Spreadsheet View
Keyboard shortcut: Ctrl‐Shift‐S (Mac OS: Cmd‐Shift‐S)
This menu item allows you to show or hide the Spreadsheet View.
Auxiliary Objects
This menu item allows you to show or hide auxiliary objects in the Algebra View.
Horizontal Splitting
This menu item allows you to split the GeoGebra window either vertically or horizontally
into the different views.
Input Bar
This menu item allows you to show or hide the Input Bar at the bottom of the GeoGebra
window.
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Command List
This menu item allows you to show or hide the list of commands next to the Input Bar at the
bottom of the GeoGebra window.
Construction Protocol…
This menu item allows you to open the Construction Protocol in a new window.
Navigation Bar for Construction Steps
This menu item allows you to show or hide the Navigation Bar at the bottom of the Graphics
View.
Refresh Views
Keyboard shortcut: Ctrl‐F (Mac OS: Cmd‐F)
This menu item allows you to repaint all views on screen.
Note: You can use this menu item to delete any traces of points or lines in the Graphics
View.
Recompute All Objects
Keyboard shortcut: F9
This menu item recomputes all objects used in your GeoGebra file.
Note: You can use this menu item to create new random numbers if you used any in your
GeoGebra file.
4.4. Options Menu
Global options may be changed in the menu Options.
Note: To change object settings, please use the Context Menu and Properties Dialog.
Point Capturing
This menu item determines if the point capturing is On or Off or if points are captured by
the grid (item On (Grid)).
Note: Option Automatic turns the point capturing On when the grid or the coordinate
system are shown and turns it Off if they are hidden.
Angle Unit
This menu item determines whether angles are displayed in Degree (°) or Radians (rad).
Note: Input is always possible in both ways (degree and radian).
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Rounding
This menu item allows you to set the number of decimal places or significant figures
displayed on screen.
Continuity
GeoGebra allows you to turn the continuity heuristic On or Off in the Options menu. The
software uses a near‐to‐heuristic to keep moving intersection points (e. g., line‐conic, conic‐
conic) close to their old positions and avoid jumping intersection points.
Note: By default this heuristic is turned off. For user defined tools continuity is always
turned off as well.
Point Style
This menu item determines whether by default points are displayed as ● dots, ○ circles, or x
crosses.
Checkbox Size
This menu item allows you to set the size of checkboxes to Regular or Large.
Note: If you are using GeoGebra as a presentation tool or are working with an interactive
whiteboard, setting the checkbox size to Large might make it easier for you to use
checkboxes.
Right Angle Style
Determines whether right angles are displayed as a rectangle □, a dot •, or like all other
angles (Off).
Coordinates
This menu item determines whether coordinates of points are displayed as A = (x, y) or
A(x | y).
Labeling
You can specify whether the label of a newly created object should be shown or not. You
can choose between the settings All New Objects, No New Objects, New Points Only, and
Automatic.
Note: The setting Automatic shows the labels of newly created objects if the Algebra View is
shown.
Font Size
This menu item determines the font size for labels and text in points (pt).
Note: If you are using GeoGebra as a presentation tool, increasing the font size makes it
easier for your audience to read text, labels, and algebraic input you are using.
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Language
GeoGebra is multilingual and allows you to change the current language setting. This affects
all input including command names and all output.
Note: No matter which language was selected, the globe icon will lead you back to the
language menu. All language names are always displayed in English.
Drawing Pad
This menu item opens a dialog where properties of the Drawing Pad (e. g. coordinate grid
and axes, background color) can be set.
Note: You can also open this dialog window by right clicking (Mac OS: Ctrl‐click) on the
drawing pad.
Save Settings
GeoGebra remembers your favorite settings (e. g., settings in the Options menu, current
Toolbar and Graphics View settings) if you select Save settings in the Options menu.
Restore Default Settings
You can restore the default settings of GeoGebra using this menu item.
4.5. Tools Menu
Create New Tool…
Based on an existing construction you can create your own tools in GeoGebra. After
preparing the construction of your tool, choose Create new tool in the Tools menu. In the
appearing dialog you can specify the output and input objects of your tool and choose
names for the Toolbar icon and corresponding command.
Note: Your tool can be used both with the mouse and as a command in the Input Bar. All
tools are automatically saved in your GGB construction file.
Manage Tools…
Using the Manage tools dialog you can delete a tool or modify its name and icon. You can
also save selected tools to a GeoGebra Tools File (GGT). This file can be used later on (File
menu, Open) to load the tools into another construction.
Note: Opening a GGT file doesn't change your current construction, but opening a GGB file
does.
Customize Toolbar…
You can customize the tools in GeoGebra's Toolbar by selecting Customize Toolbar in the
menu Tools. This is especially useful for Dynamic Worksheets where you want to restrict the
available tools in the Toolbar.
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Note: The current Toolbar setting is saved with your construction in a GGB file.
4.6. Window Menu
New Window
Keyboard shortcut: Ctrl‐N (Mac OS: Cmd‐N)
This menu item opens a new GeoGebra window that uses the default settings of the
GeoGebra interface.
Note: If you change and save some of these settings the new GeoGebra window will open
using your customized settings.
List of Open GeoGebra Windows
If you have more than one GeoGebra window open, this menu item allows you to switch
between these different windows.
Note: This might be helpful when you are using GeoGebra as a presentation tool and want
to have several GeoGebra files open at the same time as well as to toggle between them.
4.7. Help Menu
Help
This menu item gives you access to the HTML‐version of the GeoGebra Help document.
Depending on how you installed GeoGebra on your computer you may need Internet access
for this feature:
• If you downloaded and installed GeoGebra using the installer file from the webpage,
you don't need Internet access in order to use the help feature. The HTML‐version of
the GeoGebra Help document is locally saved on your computer when you install
GeoGebra.
• If you used GeoGebraWebstart to install GeoGebra on your computer, you need to
have Internet access in order to use this help feature. If you don't have Internet
access, you will get an error message.
Note: The HTML‐version of this GeoGebra Help document is available online at
If you have Internet access, this menu item opens the GeoGebra webpage in your default
web browser (
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GeoGebra Forum
If you have Internet access, this menu item opens the GeoGebra online User Forum in your
default web browser (
Note: You can post and answer GeoGebra‐related questions and problems in the GeoGebra
User Forum.
GeoGebraWiki
If you have Internet access, this menu item opens the GeoGebraWiki webpage in your
default web browser (
Note: The GeoGebra Wiki is a pool of free instructional materials created with GeoGebra by
users from all around the world.
About / License
This menu item opens a dialog window that gives you information about the license of
GeoGebra and gives credit to people who support the GeoGebra project by contributing in
many different ways (e. g., programming, translations).
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5. Special GeoGebra Features
5.1. Animation
5.1.1. Automatic Animation
GeoGebra allows you to animate one or several free numbers and/or angles at the same
time if they are shown as sliders in the Graphics View.
If you want to animate a free number or angle in GeoGebra, you need to right click (Mac OS:
Ctrl‐click) on the number or angle and select Animation On from the appearing Context
Menu. In order to stop the animation, you need to un‐check Animation On in the same
Context Menu.
Note: After animating a free number or a free angle, an animation button appears in the
lower left corner of the Graphics View. It allows you to either pause or continue an
animation.
In the Properties Dialog on tab Slider you can change the behavior of the animation:
On the one hand, you may control the Speed of the animation.
Note: A speed of 1 means that the animation takes about 10 seconds to run once through
the interval of the slider.
On the other hand, you may change how the animation cycle is repeated:
• <=> Oscillating:
The animation cycle alternates between Decreasing and Increasing.
• => Increasing:
The slider value is always increasing. After reaching the maximum value of the slider,
it jumps back to the minimum value and continues the animation.
• <= Decreasing:
The slider value is always decreasing. After reaching the minimum value of the slider,
it jumps back to the maximum value and continues the animation.
Note: While an automatic animation is activated, GeoGebra remains fully functional. This
allows you to make changes to your construction while the animation is playing.
5.1.2. Manual Animation
To manually change a number or angle continuously, select the Move tool. Then, click on
a free number or angle and press either the + or – key or the arrow keys on you keyboard.
Keeping one of these keys pressed allows you to produce manual animations.
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Example: If a point's coordinates depend on a number k like in P = (2 k, k), the point will
move along a straight line when k is changed continuously.
Note: You can adjust the increment of the slider on tab Slider of the Properties Dialog of this
object.
Keyboard Shortcuts:
• Shift + arrow key gives you a step width of 0.1 units
• Ctrl + arrow key gives you a step width of 10 units
• Alt + arrow key gives you a step width of 100 units
Note: A point on a line can also be moved along its line using the + or – key.
5.2. Conditional Visibility
Apart from just showing or hiding certain objects you can also have their visibility status
depend on a certain condition. For example, you would like an object to appear on screen if
you check a checkbox positioned in the Graphics View or if a slider is changed to a certain
value.
Conditionally Show or Hide Existing Objects
You can use the tool Checkbox to Show / Hide Objects in order to create a checkbox that
controls the visibility of one or more existing objects on screen. Alternatively, you could also
create a Boolean variable (e. g., b = true) using the Input Bar and make it visible as a
checkbox in the Graphics View by changing its visibility status (e. g., use tool Show / Hide
Object or use the Context Menu). In order to use this Boolean variable as a condition for the
visibility of certain objects, you need to follow the steps described below.
Changing the Visibility of Newly Created Objects
In the Properties Dialog, you can enter a condition for the visibility of an object on tab
Advanced.
Note: You can select the logic operators (e. g., ≠, ≥, ˄, ǁ) from the drop down menu in order
to create your conditional statements.
Examples:
• If a is a slider, then the conditional statement a < 2 means that the corresponding
object is only shown in the Graphics View if the slider's value is less than 2.
• If b is a Boolean variable, you can use b as a conditional statement. The
corresponding object is shown whenever the value of b is true and is hidden when
the value of b is false.
• If g and h are two lines and you would like a text to be shown whenever these lines
are parallel, then you could use g h as a conditional statement for the text.
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5.3. User Defined Tools
GeoGebra allows you to create your own construction tools based on an existing
construction. Once created, your custom tool can be used both with the mouse and as a
command in the Input Bar. All tools are automatically saved in your GeoGebra file.
Creating a User Defined Tool
First, create the construction your tool should be able to create later on. In the Tools menu,
click on Create New Tool in order to open the corresponding dialog box. Now you need to
fill in the three tabs Output Objects, Input Objects, and Name and Icon in order to create
your custom tool.
Example:
Create a Square‐tool that creates a square whenever you click on two existing points or on
two empty spots in the Graphics View.
• Construct a square starting with two points A and B. Construct the other vertices and
connect them with the tool Polygon to get the square poly1.
• Select Create New Tool in the Tools menu.
• Specify the Output Objects: Click on the square or select it from the drop down
menu. Also, specify the edges of the square as Output Objects.
• Specify the Input Objects: GeoGebra automatically specifies the Input Objects for you
(here: points A and B). You can also modify the selection of input objects using the
drop down menu or by clicking on them in your construction.
• Specify the Tool Name and Command Name for your new tool.
Note: The Tool Name will appear in GeoGebra's Toolbar, while the Command Name
can be used in GeoGebra's Input Bar.
• You may also enter text to be shown in the Toolbar Help.
• You can also choose an image from you computer for the Toolbar icon. GeoGebra
resizes your image automatically to fit on a Toolbar button.
Saving a User Defined Tool
You can save your custom tools so you can reuse them in other GeoGebra constructions. In
the Tools menu, select Manage Tools. Then, select the custom tool you want to save
from the appearing list. Click on button Save As… in order to save your custom tool on your
computer.
Note: User defined tools are saved as files with the file name extension GGT so you can
distinguish custom tool files from usual GeoGebra files (GGB).
Accessing a User Defined Tool
If you open a new GeoGebra interface using item New from the File menu, after you created
a custom tool, it will still be part of the GeoGebra Toolbar. However, if you open a new
GeoGebra window (item New Window from the File menu), or open GeoGebra on
another day, your custom tools won't be part of the Toolbar any more.
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There are different ways of making sure that your user defined tools are displayed in the
Toolbar of a new GeoGebra window:
• After creating a new user defined tool you can save your settings using item Save
Settings from the Options menu. From now on, your customized tool will be part of
the GeoGebra Toolbar.
Note: You can remove the custom tool from the Toolbar after opening item
Customize Toolbar… from the Tools menu. Then, select your custom tool from the
list of tools on the left hand side of the appearing dialog window and click button
Remove >. Don't forget to save your settings after removing the custom tool.
• After saving your custom tool on your computer (as a GGT file), you can import it
into a new GeoGebra window at any time. Just select item Open from the File
menu and open the file of your custom tool.
Note: Opening a GeoGebra tool file (GGT) in GeoGebra doesn't affect your current
construction. It only makes this tool part of the current GeoGebra Toolbar.
5.4. Dynamic Colors
In GeoGebra, you can change the color of objects using tab Color of the Properties Dialog.
However, you can also have the color of an object change dynamically: Open the Properties
Dialog for a certain object whose color you would like to change and click on tab Advanced.
There you will find a section called Dynamic Colors with text boxes for the color components
Red, Green, and Blue.
Note: In each of these text boxes, you can enter a function with range [0, 1].
Example:
• Create three sliders a, b, and c with an interval from 0 to 1.
• Create a polygon whose color should be influenced by the slider values.
• Open the Properties Dialog for the polygon poly1 and enter the names of the three
sliders into the text boxes for the color components.
• Close the Properties Dialog and change the values of the sliders in order to find out
how each color component influences the resulting color of the polygon.
Note: You could also animate the sliders with different speeds in order to see the
color of the polygon change automatically.
5.5. JavaScript Interface
Note: GeoGebra's JavaScript interface is interesting for users who have some experience in
HTML editing.
In order to enhance your Dynamic Worksheets and increase their interactivity, GeoGebra
applets provide a JavaScript interface. For example, you could create a button to randomly
generate new configurations of a dynamic construction.
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5.7. Labels and Captions
Show and Hide Labels
You can show or hide the labels of objects in the Graphics View in different ways:
• Select the tool Show / Hide Label and click on the object whose label you would
like to show or hide.
• Open the Context Menu for the desired object and select Show Label.
• Open the Properties Dialog for the desired object and check or un‐check the
checkbox Show Label on tab Basic.
Name and Value
In GeoGebra, every object has a unique name that can be used to label the object in the
Graphics View. In addition, an object can also be labeled using its value or its name and
value. You can change this label setting in the Properties Dialog on tab Basic by selecting the
corresponding option Name, Value, or Name & Value from the drop down menu next to the
checkbox Show Label.
Note: The value of a point is his coordinates, while the value of a function is its equation.
Caption
However, sometimes you might want to give several objects the same label, for example, to
label the four edges of a square a. In this case, GeoGebra offers captions for all objects in
addition to the three labeling options mentioned above. You can set the caption of an object
on tab Basic of the Properties Dialog by entering the desired caption into the text field called
Caption. Afterwards, you can select the labeling option Caption from the drop down menu
next to the checkbox Show Label.
5.8. Layers
Note: In GeoGebra, layers are used to determine which object to select or drag when the
user clicks on multiple objects at the same time.
By default, all objects are drawn on layer 0, which is basically the background layer of the
Graphics View. A total of 10 layers are available (numbers 0 to 9) and higher numbered
layers are drawn on top of lower numbered layers.
Using the Advanced tab of the Properties Dialog, you can change the layer for a certain
object (layers from 0 to 9 available). Once you change the layer number for at least one
object to be different from layer 0 (e. g., to layer 3), all new objects will be drawn on the
layer with the highest number that is used for any object.
Note: After selecting any object, you can select all objects in the same layer by selecting
item Select Current Layer (keyboard shortcut: Ctrl‐L) from the Edit menu. This menu item is
only available if all selected objects lie on the same layer.
87
Further use of layers:
• For SVG export objects are grouped by layer.
• Layers can be controlled using the JavaScript Interface for GeoGebra applets.
5.9. Redefine
Redefining objects is a very versatile tool to change a construction. Please note that this may
also change the order of the construction steps in the Construction Protocol.
In GeoGebra, an object may be redefined in different ways:
• Select the Move tool and double click on any object in the Algebra View.
o For free objects an editing field is opened allowing you to directly change the
algebraic representation of the object. Hit the Enter‐key in order to apply
these changes.
o For dependent objects the Redefine dialog is opened allowing you to redefine
the object.
• Select the Move tool and double click on any object in the Graphics View. This
opens the Redefine dialog and allows you to redefine the object.
• Change any object by entering its name and the new definition into the Input Bar.
• Open the Properties Dialog and change the definition of an object on tab Basic.
Note: Fixed objects cannot be redefined. In order to redefine a fixed object, you need to
free it first using tab Basic of the Properties Dialog.
Examples:
• In order to place a free point A on an existing line h, you first need to double click on
the point A to open the Redefine dialog window. Then, enter the command
Point[h] in the appearing text field and press the Enter‐key. To remove point A
from this line and make it free again, you need to redefine it to some free
coordinates like (1, 2).
• Another example is the conversion of a line h through two points A and B into a
segment. Open the Redefine dialog for line h and enter the command
Segment[A, B] in the appearing text field.
5.10. Trace and Locus
Objects can leave a trace in the Graphics View when they are moved. Use the Context Menu
to switch this Trace On. Then, modify the construction so that the object whose trace you
turned on changes its position and leaves a trace.
Note: You can turn off the trace of an object by un‐checking Trace On in the Context Menu.
The menu item Refresh Views in the View menu clears all traces.
88
You can also let GeoGebra automatically create the locus of a point by either using tool
Locus with the mouse, or enter the command Locus into the Input field.
Note: The point whose locus you would like to create must depend on another point's
movement, which is restricted to move along an object (e. g., line, segment, circle).
Example:
• Create a segment a between the points A = (‐1, ‐1) and B = (1, ‐1).
• Place a point C on the segment, so it is restricted to move along segment a.
• Create a point P that depends on point C (e. g., P = (x(C), x(C)^2)).
• Use either tool or command Locus in order to create the locus of point P in
dependence on point C:
o Tool Locus: Click first on point P and then on point C.
o Command Locus: Enter Locus[P, C] into the Input Bar and hit the Enter‐
key.
• Note: The locus created in this example is the graph of a parabola on the interval
[‐1, 1].
89 |
Before each semester, professors at Central Michigan University build a curriculum and assign textbooks to supplement lecture material. Many decide to utilize their own writing, penning their own textbooks for various assigned coursework.
Along with other benefits, the option allows instructors to design classes around textbooks they are intimately familiar with. Mathematics professor Sidney Graham praised the advantages of using his self-written course pack for his MTH 332: Introduction to Mathematical Proof course.
Generally speaking, Graham said lower level sections are assigned textbooks by a committee selection. For 200 level courses and up, professors are given the freedom to choose their own textbook.
Graham said he had been teaching classes for several years and began to notice he was using the textbook less and less each year, as he found the practice problems insufficient. He began to incorporate his own ideas into the class and eventually decided to write a course pack that would be more in line with his own approach.
"You have your own vision of what the course should be," Graham said. "I was able to write the course so that each section breaks up nicely alongside the textbook."
Lansing senior Tyler Wippel, a student of Graham's, found this to be beneficial for his competency in the course. He said since it was designed to run parallel to the progression of the class, what he was learning could be applied directly to problems in the textbook.
"It was a good book because it introduced the concepts before the math jargon like in a lot of textbooks," Wippel said. "Often they start with a lot of symbols and concepts that you don't understand until later, but this was the opposite."
Graham also uses a paperbound version to minimize the cost for his students. He said this served a dual purpose, as it allowed him to easily make revisions each semester.
Although the strategy works for Graham, such synergy between reading material and class instruction isn't always the case.
Fowlerville senior Amanda Phillips has had experiences in the past with poorly-written textbook assignments.
"The textbook was poorly written and not relevant to the course," Phillips said. "The book was very biased, and I do not believe it followed along with the course – often it would have nothing to do with what we were talking about in class."
While professors do not profit off of royalties incurred from the sales of their textbook at CMU, classroom utilization adds to their notoriety as an author.
"Overall, its their research, so it should be beneficial as long as they consider the cost and weigh their interest versus the students' interest," Wippel said.
Phillips shared a similar sentiment of the professors' responsibility to their students.
"If the course is going to be strictly off the book and the material is covered then its fine," Phillips said. "Otherwise I believe its just a way to say that many people are buying their book when otherwise their book wouldn't sell many copies."
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One Comment
Yeah…nothing better than the professor who claims that they are an accomplished author, comparing themselves with some of the biggest literary geniuses who didn't get the luxury of forcing people to buy their work.
And with those same profs….don't disagree with their theories or interpret them different…you know…thinking…or you're going to get the hammer.
If you're going to force people to buy your book to get the numbers then let the students pick their grade so they get something out of it while you make a run out of getting paid for future speaking engagements |
Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
9780471476023
ISBN:
0471476021
Pub Date: 2005 Publisher: John Wiley & Sons Inc
Summary: Did you know that games and puzzles have given birth to many of today's deepest mathematical subjects? Now, with Douglas Ensley and Winston Crawley's Introduction to Discrete Mathematics, you can explore mathematical writing, abstract structures, counting, discrete probability, and graph theory, through games, puzzles, patterns, magic tricks, and real-world problems. You will discover how new mathematical topics can ...be applied to everyday situations, learn how to work with proofs, and develop your problem-solving skills along the way. Online applications help improve your mathematical reasoning. Highly intriguing, interactive Flash-based applications illustrate key mathematical concepts and help you develop your ability to reason mathematically, solve problems, and work with proofs. Explore More icons in the text direct you to online activities at Improve your grade with the Student Solutions Manual. A supplementary Student Solutions Manual contains more detailed solutions to selected exercises in the text.
Ensley, Douglas E. is the author of Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games, published 2005 under ISBN 9780471476023 and 0471476021. Five hundred sixteen Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games textbooks are available for sale on ValoreBooks.com, one hundred four used from the cheapest price of $129.01, or buy new starting at $173.41.[read more1476023
ISBN:0471476021
Pub Date:2005 Publisher:John Wiley & Sons Inc
Valore Books is the best place for cheap Discrete Mathematics Mathematical Reasoning and Proof with Puzzles, Patterns, and Games rentals, or used and new condition books that can be mailed to you in no time. |
Mathematics and statistics
Support and resources are available for a range of courses, including mathematics, statistics, physics, chemistry, medicine and economics. You can get help with mathematics or statistics at any level of your degree program.
Learning advisers who specialise in maths are available to help you develop your skills and knowledge.
Specific maths resources and sessions can be found under the Help or Support tab in your course Blackboard site.
Small group and individual consultations can also be organised to meet your needs. Email us your name, course and subject and your query.
Maths Diagnostic Quiz- MDQ
A Maths Diagnostic Quiz is a self-diagnostic test that allows you to
measure your competence in mathematical areas such as basic algebraic
manipulations, functions, geometry and solving equations.
A number of quizzes are
available to all students via 'My Other Sites' in Blackboard, under 'Maths
Diagnostic Quiz' [ORG.106309].
You can take
a quiz at any time and immediate feedback will be provided. These quizzes should
help you to:
assess your maths ability at a basic level
determine if you have the necessary skills to study courses with maths
content
find the right person to contact if you have questions or need assistance
work out if you need to adjust your program to allow more time to build
maths skills. |
Algebra for College Students -With CD - 8th edition
Summary: Kau m...show moreathematics courses in an easy-to-read format.
TheNew to the Edition
Sections 7.1 and 7.2 have been reorganized, based on reviewer feedback. Section 7.1, Rectangular Coordinate System and Linear Equations, includes only linear equations in two variables. Section 7.2, Graphing Nonlinear Equations, emphasizes graphing nonlinear equations.
Based on reviewer feedback, the traditional concept of "series" has been replaced with the more modern idea of "finding the sum of a number of terms of a sequence" within Chapter 14. Chapter 14 has been renamed "Sequences and Mathematical Induction."
iLrn with HOMEWORK FUNCTIONALITY. Assigned by the instructor, the enhanced iLrn functionality provides direct tutorial assistance to students solving specific questions from the textbook's Problem Sets. This powerful program gives students the opportunity to try similar, algorithmically-generated problems, detailed tutorial help, the ability to solve the problem in steps, and helpful hints in solving the problem.
Updated and new problems have been integrated into this edition.
40 HOURS/WEEK OF ONLINE TUTORING are available to your students with the purchase of a new textbook. Students can access the live online vMentor tutoring service through iLrn Tutorial which is packaged with their text.
Features
THOUGHTS INTO WORDS are included in every problem set except the review exercises. These problems are designed to give students an opportunity to express in written form their thoughts about various mathematical ideas.
FURTHER INVESTIGATIONS are included in many of the problem sets. These problems offer more challenging exercises. They also provide the instructor with flexibility in incorporating advanced mathematical topics depending on specific course needs.
CLEAR-CUT 3-STEP PROBLEM-SOLVING APPROACH. Kaufmann/Schwitters straightforward, 3-step problem-solving approach is introduced and reinforced throughout the textbook. The steps are: learn the algebraic skill; use the skill to help solve equations; apply the skill to solve the application problem.
CUMULATIVE REVIEW PROBLEM SETS are included after Chapters 3, 5, 7, and 9. These help students remember skills that were introduced earlier.
PROBLEM SETS have been constructed on an even-odd basis; all variations of skill-development exercises are contained in both the even and odd numbered problems.
The Student Solutions Manual has been written by the text author, Karen Schwitters. This helps with continuity, and provides content that is consistent with the book.
CHAPTER OPENING VIGNETTES. These favorite features set the key mathematical concept in the context of the real world.
Polynomials: Sums and Differences Products and Quotients of Monomials Multiplying Polynomials Factoring: Use of the Distributive Property Factoring: Difference of Two Squares and Sum or Difference of Two Cubes Factoring Trinomials Equations and Problem Solving Summary Review Problem Set Test Cumulative Review Problem Set (Chapters 1-3)
Rectangular Coordinate System and Linear Equations Graphing Nonlinear Equations Linear Inequalities in Two Variables Distance and Slope Determining the Equation of a Line Summary Review Problem Set Test |
Stewart, the author of the worldwide best-selling calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this book to address a problem they frequently saw in their calculus courses. Many students were not prepared to "think mathematically" but attempted to memorize facts and mimic examples. ALGEBRA AND TRIGONOMETRY was designed specifically to help readers learn to think mathematically and to develop true problem-solving skills. Patient, clear, and accurate, the text consistently illustrates how useful and applicable mathematics is to real life. The new book follows the successful approach taken in the authors' previous books, COLLEGE ALGEBRA, Third Edition, and PRECALCULUS, Third Edition. |
This is the second edition of the Birkhäuser edition of 1987 that has been given a full makeover. It is a collection of papers by different authors about the definitions and descriptions and how to become familiar with polyhedra by actually building them, about their history, their role in nature and art, but also about the mathematics that are involved.
URL for publisher, author, or book:
MSC main category:
51 Geometry
MSC category:
51M20
Other MSC categories:
90C57, 52B20, 68R10, 97K30, 00A99
Review:
As the editor quotes in her introduction "plus que ça change, plus c'est la même chose" since indeed polyhedra are as new as they are old and given the recent evolution in graphs, discrete and computational geometry, combinatorial optimization, computer graphics, a new edition of the previous version (Birkhäuser, 1987) became unavoidable and it resulted in a complete makeover. The format is still the same (the first edition grew out of a 1984 conference), it consists of a collection of essays by different authors about many different aspects related to polyhedra.
The papers are ordered in such a way that they start with elementary, less formal definitions an properties, and suggestions and practical tips about how to actually organize hands-on sessions where children are encouraged to construct the three-dimensional objects. But polyhedra are also followed along their historical and cultural trail from the pyramids in old Egypt and the Platonic solids, till recent developments.
In a second part, appearance and use of polyhedra in art and nature is the the central theme. They lived in the minds of the architects of the pyramids but they also appear in futuristic constructions of modern architecture. Because their graphs have some optimality and stability properties also nature's architect is eager to make use of these structures. Crystals, chemical bindings, cell biology quite often follow the geometrical laws of polyhedral constellations. And of course many artists made 2 of 3-dimensional artwork inspired by these forms.
In part 3, called "polyhedra in geometrical imagination", the contributions become more mathematical. Here we find more general polyhedra, and discussions about molecular stability, dual graphs, Dirichlet tessellations and spider webs, diophantine equations, rigidity, decomposition of solids, etc. The final contribution is a set of 10 geometrical problems that are still (partly) open problems still waiting for a solution.
Although there are 22 papers by many different authors, there is an extensive global index that helps you to find the items you are looking for. The readability of the papers is kept as smooth as possible by collecting notes, remarks and references in a section at the end of the book. Of course the style cannot be uniform since there is a difference between an historical survey, an exposition of how to glue pieces of cardboard together, and a mathematical paper with theorems. However, by the ordering of the papers, the reader grows gradually into the mathematics as he of she is reading on towards the end of the book.
The book is amply illustrated and aiming at a public from 9 till 99. It will be of interest to a very broad public. Form a mathematical side children might be interested in geometrical puzzles and advanced mathematicians may be interested in solving the open problems, and the whole range in between will probably find something interesting of their own taste. But of cause also the non-mathematician will be attracted by these fascinating building blocks in nature, art, science and engineering.
Reviewer:
A. Bultheel
Affiliation:
KU Leuven
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A program to findprimeCalculate the prime numbers between two limits. It uses a deterministic test and uses the Sieve of Eratophenes logic. Described on The Math Forum @ Drexel. Recommended for Middle School, High School and College, ages 6+. 99¢ only. Educational discount: 20%.MathProf is an easy to use mathematics program within approximately 180 subroutines. MathProf can display mathematical correlations in a very clear and simple way. The program covers the areas Analysis, Geometry, Algebra, Stochastics,Vector algebra.MathProf helps Junior High School students with problems in Geometry and Algebra. High School and College students, seeking to expand their knowledge into further reaching mathematical concepts find thSim card contacts reader program recovers lost contactnumber |
Find a La Marque ScienceDiscrete math is normally divided into six areas: sets, functions, and relations; basic logic; proof techniques; counting basics; graphs and trees; and discrete probability. I show students how these topics are interwoven with computer science applications. I also help students use the tools of discrete math to see computational problems from a mathematical perspective. |
...But My goal is to make whatever is challenging you most seem easyA typical Algebra 1 course reinforces the very basics of solving, graphing, and writing linear equations and inequalities. The next step introduces powers and exponents, quadratic equations along with polynomials and factoring. I like to emphasize WHY a particular concept is being learned along with the real-life CONNECTION. |
89FUNDAMENTALS OF MATHEMATICS3 UnitsIntroduces basic skills mathematics for students with disabilities. Uses special instructional methods and materials to teach learning strategies for computation, problem solving, and real life applications. Prepares students to enter into MATH 20 or Career Technical Certificate Programs. [ND]
PD 89 Sections Offered
Cls No.
Crs/Sect No.
Time
Days
Room
Instructor
#Weeks
#Seats
85223
PD 89 S01
11:00AM-12:15PM
MW
436
Charter,Patricia
19
6*
MAIN CAMPUS - DAY
Begins: 1/21/2014 Ends: 5/30/2014This course is designed for students with disabilities.*Open seats as of Friday, April 11, 2014 8:14 AM |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more |
Using the author's considerable experience of applying Mathcad to engineering problems, Essential Mathcad introduces the most powerful functions and features of the software and teaches how to apply these to create comprehensive calculations for any quantitative subject. The simple, step-by-step approach makes this book an ideal Mathcad text for professional engineers as well as engineering , science, and math students. Examples from a variety of fields demonstrate the power and utility of Mathcad's tools, while also demonstrating how other software, such as Excel spreadsheets, can be incorporated effectively. A companion CD-ROM contains a full non-expiring version of Mathcad 14 (North America only). The included software is for educational purposes only.
Mathcad is the industry-standard software for engineering calculations. Its easy-to-use, unitsaware, live mathematical notation, powerful capabilities, and open architecture allow engineers and organizations to streamline critical design processes.
Description: The most powerful scientific package for data analysis and processing. It is considered the de facto standard in many research laboratories. OriginPro is a universal means of statistical and mathematical functions, handling data files, charting and graphs of these functions on the data arrays, as well as the development tool specialized mathematical software and visualization of image data. Integration with data collection systems such as LabView, DasyLab, LabWindows ... Compatible with MathLab, MathCad, Microsoft Office ...
MATLAB - a high-level technical computing language, interactive environment for algorithm development and modern tools of data analysis. Also present in the assembly library of books on the package MATLAB & Simulink. |
Math Resources and Portals
Matematicas Visuales | Home
In MatematicasVisuales you will find visual expositions of mathematical concepts. MatematicasVisuales intends to complement the work initiated by artiludios , a site with games, puzzles and mathematical curiosities . Reading Miguel de Guzmán I found a demonstration of the line of Simpson and the Steiner Deltoid . It serves as an introduction to the geometry section. The concept of function and its graphical representation are a key concept and we dedicate special attention to it in the analysis section. Geometric representation of the complex numbers facilitates its visualization.
CR Algebra Review
Prealgebra Review A review of the concepts in Prealgebra as preparation to enter Elementary Algebra (Math 380). Elementary Algebra Review A review of the concepts in Elementary Algebra as preparation to enter Intermediate Algebra (Math 120). Intermediate Algebra Review A review of the concepts in Intermediate Algebra as preparation to enter a Transfer Level math class (Math 5, 15, 25, or 30). Important Information Procedure: Each review course is broken up into 6 modules. Each module has 2 or 3 skills (labeled A, B, ...) to be reviewed.
Insight Maker runs in your web-browser. No software download or plug-ins are needed. Get started building your rich pictures, simulation models and Insights now. Features Explore powerful simulation algorithms for System Dynamics and Agent Based Modeling in a truly cohesive environment.
Math Fortress | College Math Video | Math Video | Equation Sheets | Math Work Sheets | Math Resources | Algebra | Geometry | Calculus | Linear Algebra | Differential Equations | GRE
Welcome to MathFortress.com where you can "Fortify Your Math Knowledge". Here you will find unique Great Quality Videos and other resources for various mathematical disciplines all for FREE! You will need Adobe Flash Player to see the videos. This website is best viewed using Chrome or Firefox browsers.
Project MATHEMATICS!
DVDs available in English & Spanish Project description Project MATHEMATICS! videos explore basic topics in high school mathematics in ways that cannot be done at the chalkboard or in a textbook.
Mathematical Formula Tables
The following links are to the Mathematical Formula Tables produced by the Mathematics Department. This gives the same information as the pink-covered booklet and the yellow-covered booklet available in exams. Currently the tables are available in and pdf format. These can be downloaded and/or viewed with Adobe Acrobat. You can access the individual sections or the tables as a whole.
Welcome to AlgebraLAB, an online learning environment that focuses on topics and skills from high school mathematics that students must be able to draw upon in their introductory science courses. Since math is the language of science, science courses are often where students first realize "Oh, so this is why we learned that in algebra …" In many cases students often discover that it is one or more math skills that initially block their ability to understand and internalize new science concepts. In science classes students must learn how to recognize when particular mathematics procedures are applicable so that they can select from their "mathematics toolboxes" the correct methods needed to solve new problems.
Making Math and Science Connections
Fill-It-In Outline Mathematics
What about free graphing and algebra on your mobile phone? XCalc is a free Java Applet which offers a graphing calculator for any mobile phone which runs Java (which is nearly all of them now!) Think about the possibilities for classes that do not yet have access to graphing calculators! While limited (it does not yet have a trace facility) it DOES offer 2d and 3d graphing which is quite fast and powerful.
Free Mathematics Resources OnLine
Graphers
Mathematics Assessment
Authentic Assessment in Mathematics Home Page
The Geometry Forum Summer '94 Workshop at Swarthmore College, Swarthmore, PA. The goal of this project was to collect and organize available INTERNET resources on Authentic Mathematical Assessment applicable to Secondary Schools. What is it?
Go to my home page Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com. "He must be a 'practical' man who can see no poetry in mathematics." - W.
Jim Loy's Mathematics Page |
This set allows for multiple students who wish to use Teaching Textbooks: Geometry. Perfect for siblings, co-ops or advanced students, this set contains a 768 page softcover, spiralbound student textbook and 164 test book with answer key. The audiovisual lecture, practice, and solution CDs are NOT included.
Good if you just want a review
Date:August 22, 2012
Casey
Location:Southwest USA
Age:45-54
Gender:female
Quality:
2out of5
Value:
4out of5
Meets Expectations:
3out of5
NEED TO KNOW:The authors assume you already know certain skills, such as how to do a lot of algebra. There are so many times where they leave out steps so if you are looking for a book that can teach you or your children geometry, I would not recommend this. If you just want a review and can fill in, supplement the parts they do not show then this shoud work well for you. PROS: What I do like is the practice lessons give lots of practice so if you have a child that needs a lot of repetition, the practice sections are great. Another thing I like is that the practice sections continue to review things from earlier sections so that they are not forgotten.
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-1point
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Review 2 for Teaching Textbooks Geometry Textbook with Answer Key
Overall Rating:
5out of5
Great math program
Date:July 14, 2011
Tanya Huwalt
Location:Pa. homeschooling mom of a large family. We originally tried Teaching Textbooks to help my one daughter, who was struggling with math. She was to the point of hating it. I switched her to Teaching Textbooks back in grade 4. She went to straight A's. She is now doing 7th grade Teaching Textbook. My other children, who used to watch her on the computer all loved the program and wanted to try it. Now, all of them are using the Teaching Textbook for math. I would never use another program. We had tried others & I feel this one is what works best with the many different learning styles of my kids. They all are doing very well in math.
Share this review:
+1point
1of1voted this as helpful.
Review 3 for Teaching Textbooks Geometry Textbook with Answer Key
Overall Rating:
5out of5
Very Informative
Date:July 1, 2011
Generosa Tummond
Location:South Carolina
Age:25-34
Gender:female
Quality:
5out of5
Value:
2out of5
Meets Expectations:
4out of5
I purchased this item for my son for next school year. So we haven't begun to use it yet, but it looks very detailed and thorough.
Share this review:
-2points
0of2voted this as helpful.
Review 4 for Teaching Textbooks Geometry Textbook with Answer Key
Overall Rating:
5out of5
Date:April 1, 2010
Lewis Veltema
My daughter LOVES Teaching Text Books. It makes learning math easy for her. We have tried other programs and like Teaching Test Books best. Thank you for offering such a good program. Homeschool mom |
Algebra: Word Problems Help and Practice Problems
Find study help on linear applications for algebra. Use the links below to select the specific area of linear applications you're looking for help with. Each guide comes complete with an explanation, example problems, and practice problems with solutions to help you learn linear applications for algebra.
The following exam will test your knowledge of various types of word problems. The 50 questions are presented in the same order that the topics they cover are presented in this book. If you are more comfortable with some topics than others, such as probability or ... |
Calculus Latin, calculus, a small stone used for counting is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
In American mathematics education, precalculus or Algebra 3 in some areas, an advanced form of secondary school algebra, is a foundational mathematical discipline. It is also called Introduction to Analysis. In many schools, precalculus is actually two separate courses: Algebra and Trigonometry. Precalculus prepares students for calculus the same way as pre algebra prepares students for Algebra I. While pre algebra teaches students many different fundamental algebra topics, precalculus does not involve calculus, but explores topics that will be applied in calculus. Some precalculus courses might differ with others in terms of content. For example, an honors level course might spend more time on topics such as conic sections, vectors, and other topics needed for calculus. A lower level class might focus on topics used in a wider selection of higher mathematical areas, such as matrices which are used in business.
Mathematics Describing the Real World Precalculus and Trigonometry-Bruce H. Edwards AVI, XviD, 640x480, 29.97 fps | English, MP3@128 kbps , 2 Ch | ~36x30 mins | 10.82 GB The Teaching Company | 2011 | Course no. 1005 Trad... Filesonic, Fileserve, Uploading, Wupload, Uploadstation Links Engoy all members !!!...
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With the same design and feature sets as the market leading Precalculus, 8/e, this addition to the Larson Precalculus series provides both students and instructors with sound, consistently structured explanations of the mathematical concepts. Designed for a two-term course, this text contains the features that have made Precalculus a complete solution for both students and instructors: interesting applications, cutting-edge design, and innovative technology combined with an abundance of carefully written exercises. In addition to a brief algebra review and the core precalculus topics, PRECALCULUS WITH LIMITS covers analytic geometry in three dimensions and introduces concepts covered in calculusApplied Calculus for Business, Economics, and Finance is a combination of the authors' two previous texts Precalculus and Elements of Calculus and Applied Calculus. This single text may be used to cover the content of an applied calculus course for non-science majors. Continuing the approach used in its precursor texts, Applied Calculus for Business, Economics, and Finance features the integration of precalculus with the calculus as well as the integration of technology with both subjectsDr. Jenny Switkes will help you master the intricacies of Calculus from Limits to Derivatives to Integrals. In Educator's Calculus 1 course, Professor Switkes covers all the important topics with detailed explanations and analysis of common student pitfalls. Calculus can be difficult, but Professor Switkes will show you how to reap the rewards of your hard work, all while showing you the beauty and importance of math. Whether you just need to brush up on your calculus skills or need to cram the night before the final, Professor Switkes has taught mathematics for 10+ years and knows exactly how to help. |
Mathematics
What Will You Learn? What Skills Will You Acquire?
You will learn how to reason your way through unfamiliar territory, find familiar structures, make predictions and answer important questions. Mathematics is central to science, engineering, finance, insurance and computing for precisely these reasons. You will see many mathematical structures and understand how they apply to many pursuits.
In your senior research project, you will apply your knowledge to a single challenging problem. These projects have included choosing optimal immunization strategies for a heterogeneous network, proving a matrix decomposition resulting from a multiplicative identity, using statistical mechanics to properly value financial options, and proving a relationship between offensive and defensive performance and the eventual outcome of a baseball game.
From Western's Calculus 3 course: Students built parametric curves describing ski slopes and their likely paths.
Beyond the Classroom
The Math department has one of the most active student communities on campus. As a math student, you will enjoy a pre-built learning community for your schoolwork, as well as many social and scholarly activities organized by the faculty. In the fall, we host a welcome-back barbecue with a kickball game.
Each fall the department has a barbecue to welcome our returning students back to campus and to help our new students become part of our community. Both of these events are open to all students and faculty interested in mathematics or computer science, their friends and family. In the winter, students in the department attend the Pikes Peak Regional Undergraduate Mathematics Conference where our seniors present their research projects. Some years, we have an ice-climbing outing. Others, it is a skating party. The high point of our spring social calendar is the annual MCIS banquet. The faculty caters this affair, so you will see we are not only excellent scholars and kickball players, but we can also can make a mean pan of mac 'n cheese.
But life is not all kickball and casseroles. For professional advancement, we help show you what we do and point you to opportunities. The math seminar meets at noon Monday, Wednesday or Friday. At these gatherings, faculty members show what they are working on and seniors to present their research projects. In February, we load up a couple of vans and travel to the Pikes Peak Undergraduate Research Conference, both to show off what our seniors have done and to see what others are doing.
Many of our students use the summer months to pursue internships or other advanced training. Our students have successfully landed positions in summer workshops offered by the institute for Advanced Studies and Research Experiences for Undergraduates (REUs), offered by the National Science Foundation.
After Graduation
As a Western Mathematics graduate, you will find opportunities both within the field and in other pursuits. Our graduates have earned advanced degrees in math, engineering, geology and architecture.
Our graduates are working toward master's degrees and PhDs with full financial support. If you want to teach, you will be in great demand. There are many programs that will allow you to begin teaching and complete your licensure requirements online. According to a recent survey, even our graduates who are not working in the field said they used math "almost every day" in their chosen professions. Common wisdom suggests that a math degree pays off commensurately with the level you rise to in your profession.
Courses
MATH 98 - BEGINNING ALGEBRA (3 credits)
An introduction to algebra with a review of basic arithmetic. Includes decimals, fraction, percentage, ratio, proportion, signed numbers, algebraic expressions, factoring, exponents and radicals, linear equations, and graphs. MATH 098 is offered through Extended Studies and a fee is assessed. Credit does not count toward graduation. Graded Satisfactory/Unsatisfactory only.
MATH 99 - INTERMEDIATE ALGEBRA (3 credits)
A review of the arithmetic of fractions and decimals, percentage problems, signed numbers, arithmetic, and topics of basic algebra, including simplifying algebraic expressions, solving and graphing linear equations, basic factoring, working with algebraic fractions, and solving rational and quadratic equations. This course is designed for students who need a review of the basic algebra skills necessary to complete the required mathematics courses MATH 131 or MATH 140. MATH 099 is offered through Extended Studies and a fee is assessed. Credit does not count toward graduation. Graded Satisfactory/Unsatisfactory only. Prerequisite: ACT math score of 16 or above
MATH 105 - MATH FOR LIBERAL ARTS GMA1 (3 credits)
An investigation of a number of mathematical concepts, which may include ratios and proportions, descriptive statistics, sets and logic, geometry, right-angle trigonometry, counting, and probability. A variety of teaching methods are employed such as cooperative groups, writing about mathematics, and technology (calculators and computers). Prerequisite: ACT math score of 19 or above
MATH 140 - COLLEGE ALGEBRA GMA1 (3 credits)
An integration of the essential algebraic manipulations, solving equations and inequalities, polynomial functions, exponential and logarithmic functions, and techniques of graphing. Prerequisite: ACT math score of 19 or above
MATH 141 - PRECALCULUS (4 credits)
Preparation for calculus by the study of functions of one variable over the real numbers. These are introduced in general and then applied to the usual elementary functions, namely polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions. Inverse functions, polar coordinates and trigonometric identities are included. Prerequisite: ACT math score of 23 or above
MATH 209 - MATH FOR ELEM SCHL TEACHERS I (3 credits)
First of two courses designed for prospective elementary teachers. Emphasizes the real number system, arithmetic operations, and algebra. Explorations focus on representing, analyzing, generalizing, formalizing, and communicating patterns and structures. Content is presented using problem solving and exploration. Prerequisite: ACT math score of 23 or above, SAT math score of 530 or above
MATH 213 - PROBABILITY STATISTICS GMA1 (3 credits)
An introduction to descriptive statistics, probability concepts, and inferential statistics. The topics for the course include presentation of data, counting principles, probability rules, and discrete and continuous probability distributions. Prerequisite: MATH 141 with a minimum grade of "C-,"' or Accuplacer College-Level Mathematics test score of 85 or above
MATH 220 - INTRO TO ADVANCED MATHEMATICS (3 credits)
Students develop and use elementary logic and set theory to construct deductive proofs with relations, functions, and some algebraic structures. Topics include indexing, equivalence relation theory, and cardinality. Prerequisite: MATH 151 with a minimum grade of "C-."
MATH 251 - CALCULUS II (4 credits)
Topics include techniques of integration, area computations, improper integrals, infinite series and various convergence tests, power series, Taylor's Formula, polar coordinates, and parametric curves. Prerequisite: MATH 151 with a minimum grade of "C-."
MATH 252 - CALCULUS III (4 credits)
Topics include calculus of functions of several variables, differentiation and elementary integration, vectors in the plane and space. Prerequisite: MATH 251 with a minimum grade of "C-."
MATH 266 - SEC MATH FROM AN ADV PERSPCTV (3 credits)
A course designed to help Secondary Licensure Emphasis majors understand the core mathematical content of high school mathematics courses before calculus. These concepts are treated from an advanced standpoint, emphasizing connections and extensions. Topics include number systems, polynomial and transcendental functions, analytic geometry, theory of equations, and measurement. Prerequisite: MATH 151 with a minimum grade of "C-."
MATH 300 - INTRO TO MATHEMATICAL MODELING (3 credits)
Designed to teach the basic principles of mathematical modeling and applied mathematics. Techniques from calculus, statistics, and probability are utilized to model real-world problems. Analytic and numeric tools are used to implement the models, obtain predictions and investigate underlying mechanisms. Topics include dimensional analysis, curve fitting, simulations, differential and difference equations. Prerequisites: MATH 251 and MATH 213 with minimum grades of "C-."
MATH 451 - ANALYSIS I (3 credits)
An introduction to the theory of calculus. Topics include the usual topology of the reals, sequences, limits, continuity, differentiation, and Riemann integration. Prerequisites: MATH 220 and MATH 252 with minimum grades of "C-."
MATH 495 - SENIOR SEMINAR (2 credits)
A Capstone Course for the Mathematics Standard Major and for the Secondary Licensure Emphasis. Each student selects an area of interest, researches the selected area, generates a reference list and research paper, and presents the paper to a seminar of faculty and students. Prerequisites: MATH 360 and either MATH 451 or MATH 471Dorgan, William E. Memorial Math Scholarship
Available to:
Full-time students at Western. Must be a declared major in Mathematics. Must have a GPA of 3.0 or greater.
Provided by:
The estate and family to celebrate the memory of William E. Dorgan.
Amount:
Award depends on funds available.
Selected by:
Western Math faculty in consultation with the Office of Admissions and Financial Aid as needed
Mathematical Studies Scholarship
Available to:
Students who have successfully completed a minimum of 24 credits at Western and at least 6 credits that apply toward the major. Must be a declared major in Mathematics. Must have a GPA of 3.0 or greater in Math at Western that apply toward the major. Must be enrolled for at least 9 credits at Western during the semester of application. Financial need and the student's contribution to the math program will also be considered. |
The basics of computer algebra and the language of Mathematica are described. This title will lead toward an understanding of Mathematica that allows the reader to solve problems in physics, mathematics, and chemistry. Mathematica is the most widely used system for doing mathematical calculations by computer, including symbolic and numeric calculations... more...
Focusing on robust rank-based nonparametric methods, this book covers rank-based fitting and testing for models ranging from simple location models to general linear models for uncorrelated and correlated responses. Illustrated with real data examples using R, each chapter includes a short problem set with data sets. The corresponding example codes... more theoretical and methodological aspects of MFA. It also... more...
Learn How to Program Stochastic Models
Highly recommended, the best-selling first edition of Introduction to Scientific Programming and Simulation Using R was lauded as an excellent, easy-to-read introduction with extensive examples and exercises. This second edition continues to introduce scientific programming and stochastic modelling in... more...
As one of the most popular software tools in the world, Microsoft Excel is used by business analysts across the globe to keep track of details, look deeply into the data that drives business, and provide managers with the information and insights they need to keep their companies moving forward. MICROSOFT EXCEL 2013 FOR THE BUSINESS ANALYST is an introduction... more...
This knowledge of financial theory is assumed, but familiarity with R is not required. With... more... |
Algebra II Practice Book Grades 8+
# CD404043
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Take algebra to the next level with practice problems that cover inequalities; linear equations; polynomial products and factors; rational expressions; roots, radicals, and complex numbers; quadratic equations and functions; as well as variation! Struggling students will benefit from clear instructions and examples plus the Check-Up Problems section, which assesses what they have learned. Also provided are references, tips, and answer keys. 128 pages |
Electronics and Computer Math - 8th edition
The Eighth Edition continues the tradition of providing the most complete, thorough, and practical coverage of mathematics and its application in the world of electronics. This market leading text has been used in hundreds of classrooms by thousands of students who have benefited from the clear explanations and large practice sets that develop both computational and quantitative thinking skills.
Features <...show moreBR> Core Features Carried Over From the Seventh Edition:
Key Points highlight key concepts in each section.
Chapter topics are broken into small testable sections that move students carefully from concept to concept.
Over 300 worked examples model problem solving steps for students.
Over 1400 practice problems within the chapters provide a wealth of practice problems.
Self-tests at the end of each chapter sub-topic allow students to check their understanding upon completion of a topic.
Over 3500 end-of-chapter problems including over 500 new word problems that help students think and apply concepts quantitatively.
Dust Cover Missing. Millions of satisfied customers and climbing. Thriftbooks is the name you can trust, guaranteed. Spend Less. Read More.
$114118.48 |
IGCSE Mathematics for CIE (International GCSE)
Description: Innovative - Interactive - InternationalCollins IGCSE Mathematics has been developed to give maximum support for students studying for the Cambridge International Examinations GCSE. International examples are used throughout, 'localising' learning. |
MUS Transfer
MUS Transfer Core
Transferable General Education Curriculum
Mathematics
Every day we are inundated with numerical information, often in the form of graphical representations, statistical summaries, or projections from mathematical models. Comprehension of elementary quantitative concepts, development of quantitative reasoning skills, and the ability to reasonably ascertain the implications of quantitative information are goals of mathematics courses.
Mathematics core courses will:
expose students to the methods employed in the mathematical sciences;
demonstrate the application of mathematical or statistical models to complex problems, which can lead to greater understanding of, and potential solutions to, these problems;
enable students to develop skills leading to an understanding of quantitatively-based problems of importance to contemporary society;
provide practical applications that relate to students' personal and future professional lives as consumers of quantitative information. |
Product Description
This Saxon Algebra 2 Home Study Kit includes the Student Textbook, Test Book/Answer Key Packet, and Solutions Manual. Traditional second-year algebra topics, as well as a full semester of informal geometry, are included with both real-world, abstract and interdisciplinary applications. 129 lessons cover topics such as geometric functions, angles, perimeters, and proportional segments; negative exponents; quadratic equations; metric conversions; logarithms; and advanced factoring.
Student Text is 577 pages, short answers for problem/practice sets, an index and glossary are included; hardcover.
The Homeschool Packet includes contains 32 student tests and test solutions with work shown along with the final answer. The Answer key portion shows only the final solution for the practice and problem sets found in the student text. 44 pages, paperback.
The Solutions Manual features solutions to all textbook practices and problem sets. Early solutions contain every step, while later solutions omit obvious steps. Final answers are in bold type for accurate, efficient grading.
Product Reviews
Saxon Algebra 2
I've used Saxon Math books since level 54 and love the constant review of each new concept. The level of retention is higher with the constant review while learning new concepts.
While Saxon does not teach ALL the Algebra concepts needed (85%) for College if you use the DIVE CD, like my son did. you will find an additional Bonus 20 lessons to help your child pass the CLEP exam. You'll even find sample test to make sure you're successful to pass the test.
Love the Saxon Concept, but was disappointed by the number of Typo's in the book.
October 30, 2010
This book is great if you are looking to strengthen geometry skills, but it is not on a public school level for an Algebra 2 book. After going through the book, I returned it and chose to teach from the local High School Algebra 2 book.
August 6, 2010 |
Mathematical and Computer Programming Techniques for Computer Graphics introduces the mathematics and related computer programming techniques used in Computer Graphics. Starting with the underlying mathematical ideas, it gradually leads the reader to a sufficient understanding of the detail to be able to implement libraries and programs for 2D and 3D graphics. Using lots of code examples, the reader is encouraged to explore and experiment with data and computer programs (in the C programming language) and to master the related mathematical techniques. A simple but effective set of routines are included, organised as a library, covering both 2D and 3D graphics – taking a parallel approach to mathematical theory, and showing the reader how to incorporate it into example programs. This approach both demystifies the mathematics and demonstrates its relevance to 2D and 3D computer graphics.
Table of Contents
Table of Contents
Vector Algebra Survival Kit.
Matrix Algebra Survival Kit.
Vector Spaces or Linear Spaces.
Two
Dimensional Transformations.
Two
Dimensional Clipping.
Three
Dimensional Transformations.
Viewing and Projection Transformations.
3D Rendering |
Khan Academy: Readers weigh in
Recent posts on the Khan Academy — including an e-mail I posted from founder Sal Khan — sparked a lot of interest and reaction from readers.
The Khan Academy is essentially an on-line library of more than 3,300 videos on subjects including math, physics, and history that are designed to allow students to learn at their own pace and for teachers to use as Sal Khan
(Courtesy Khan Academy)
instructional tools.
One post, titled "Khan Academy: The hype and the reality," by Karim Kai Ani, a former middle school teacher and math coach, and the founder of Mathalicious, took issue with the way Khan Academy videos deal with the concept of slope. Sal Khan sent in a response to the critique, which you can find here.
I asked readers to weigh in and following are several of the responses I received. The positions are different, showing how complex something that seems simple can be.
Here they are:
This was written Raymond Johnson and Frederick Peck, Ph.D. students in mathematics education at the University of Colorado at Boulder and the Freudenthal Institute US. Raymond and Fred each have six years of experience teaching Algebra 1 and are engaged in research on how students understand slope and linear functions. Raymond blogs about math education and policy at MathEd.net and Fred shares his research and curriculum at RMEintheClassroom.com. This post first appeared on MathEd.net.
The Answer Sheet has recently been the focus of a lively debate pitting teacher and guest blogger Karim Kai Ani against the Khan Academy's Salman Khan. While Karim's initial post focused mainly on Sal Khan's pedagogical approach, Karim also took issue with the accuracy of Khan Academy videos. As an example, he pointed to the video on "slope." Specifically, Karim claimed Sal's definition of slope as "rise over run" was a way to calculate slope, but wasn't, itself, a definition of slope. Rather, Karim argued, slope should be defined as "a rate that describes how two variables change in relation to one another." Sal promptly responded, saying Karim was incorrect, and that "slope actually is defined as change in y over change in x (or rise over run)." To bolster his case Sal referenced Wolfram Mathworld, and he encouraged Valerie Strauss to "seek out an impartial math professor" to help settle the debate. We believe that a better way to settle this would be to consult the published work of experts on slope.
Working on her dissertation in the mid-1990s, Sheryl Stump (now the Department Chairperson and a Professor of Mathematical Sciences at Ball State University) did some of the best work to date about how we define and conceive of slope. Stump found seven ways to interpret slope, including: (1) Geometric ratio, such as "rise over run" on a graph; (2) Algebraic ratio, such as "change in y over change in x"; (3) Physical property, referring to steepness; (4) Functional property, referring to the rate of change between two variables; (5) Parametric coefficient, referring to the "m" in the common equation for a line y=mx+b; (6) Trigonometric, as in the tangent of the angle of inclination; and finally (7) a Calculus conception, as in a derivative. (See below for the reference to Stump's work.)
If you compare Karim and Sal's definitions to Stump's list, you'll likely judge that while both have been correct, neither has been complete. We could stop here and declare this duel a draw, but to do so would foolishly ignore that there is much more to teaching and learning mathematics than knowing what belongs in a textbook glossary. Indeed, research suggests that a robust understanding of slope requires (a) the versatility of knowing all seven interpretations (although only the first five would be appropriate for a beginning algebra student); (b) the flexibility that comes from understanding the logical connections between the interpretations; and (c) the adaptability of knowing which interpretation best applies to a particular problem.
All seven slope interpretations are closely related and together create a cohesive whole. The problem is, it's not immediately obvious why this should be so, especially to a student who is learning about slope. For example, if slope is steepness, then why would we multiply it by x and add the y-intercept to find a y-value (i.e., as in the equation y=mx+b)? And why does "rise over run" give us steepness anyway? Indeed, is "rise over run" even a number? Students with a robust understanding of slope can answer these questions. However, Stump and others have shown that many students — even those who have memorized definitions and algorithms — cannot.
This returns us to Karim's original point: There exists better mathematics education than what we currently find in the Khan Academy. Such an education would teach slope through guided problem solving and be focused on the key concept of rate of change. These practices are recommended by researchers and organizations such as the NCTM, and lend credence to Karim's argument for conceptualizing slope primarily as a rate. However, even within this best practice, there is nuance. For instance, researchers have devoted considerable effort to understanding how students construct the concept of rate of change, and they have found, for example, that certain problem contexts elicit this understanding better than others.
Despite all we know from research, we should not be surprised that there's still no clear "right way" to teach slope. Mathematics is complicated. Teaching and learning is complicated. We should never think there will ever be a "one-size-fits-all" approach. Instead, educators should learn from research and adapt it to fit their own unique situations. When Karim described teachers on Twitter debating "whether slope should always have units," we see the kind of incremental learning and adapting that moves math education forward. These conversations become difficult when Sal declares in his rebuttal video that "it's actually ridiculous to say that slope always requires units*" and Karim's math to be "very, very, very wrong." We absolutely believe that being correct (when possible) is important, but we need to focus less on trying to win a mathematical debate and focus more on the kinds of thoughtful, challenging, and nuanced conversations that help educators understand a concept well enough to develop better curriculum and pedagogy for their students.
This kind of hard work requires careful consideration and an open conversation, even for a seemingly simple concept like slope. We encourage Sal to foster this conversation and build upon what appears to be a growing effort to make Khan Academy better. Doing so will require more than rebuttal videos that re-focus on algorithms and definitions. It will require more than teachers' snarky critiques of such videos. Let's find and encourage more ways to include people with expertise in the practice and theories of teaching mathematics, including everyone from researchers who devote their lives to understanding the nuance in learning to the "Twitter teachers" from Karim's post who engage this research and put it into practice. This is how good curriculum and pedagogy is developed, and it's the sort of work that we hope to see Sal Khan embrace in the future.
*Sal's point is that if two quantities are both measured in the same units, then the units "cancel" when the quantities are divided to find slope. As an example, he uses the case of vertical and horizontal distance, both measured in meters. The slope then has units of meters/meters, which "cancel". However, the situation is not so cut and dry, and indeed, has been considered by math educators before. For example, Judith Schwartz (1988) describes how units of lb/lb might still be a meaningful unit. Our point is not to say that one side is correct. Rather, we believe that the act of engaging in and understanding the debate is what is important, and that such a debate is cut short by declarative statements of "the right answer."
This was written by Martin Weil, a physicist who happens also to be a brilliant colleague of mine at The Washington Post who usually manages to suppress his views on major public issues.
What I would say, at the outset, however, is that if I wanted to take issue with Khan, it would not be over his definition of slope. This may add to your understanding of that narrow issue. It would not be worth going into any detail, were it not that slope and the concept of slope is at the heart of calculus.
Khan's definition of slope is a good approximation. It introduces the concept and gives an intuitive feel for it. Calculus depends on a refined version of that rough and ready idea.
In a sense, everybody knows what "slope" is. If you drive your car uphill, and you start at sea level and rise in one mile to a height of 100 feet, then the slope is 100 feet per 5280 feet....which is about 1/50. We would say that the road has a 2 per cent slope, or a two per cent grade. That is a concept that needs little explanation. If the road rises steadily from the bottom to the top, we can say that at every point in the route, the slope is 2 per cent. Or one in 50. At every point. (note that no units are required) It's the rise over the run. That's the slope
At every point of the route, you are going up a 2 per cent grade. A 2 per cent slope.
Calculus deals with more sophisticated and complicated problems. In these, the slope may change at every yard, every foot, every INCH along the way. Knowing the precise i rate at which things are changing, with time, with distance, with some other variable, makes it possible to solve a variety of significant and important problems.
The basic concept is indeed " rise over run" . Nothing wrong with that. But rise over run where? at what point on the road? Well, the example I have given is an easy one. Because rise over run is the SAME at every point. It's 100 feet per mile. 100 feet per 5280 feet. In this case of an unvarying slope, it is at every point on the road, 1 foot peer 52.8 feet. and so on.
But calculus is applied to more complicated problems. You can not solve a calculus problem by looking at it and repeating to yourself "rise over run." In calculus problems the slope changes constantly. No matter how brief the rise, how short the run, the slope differs at every point. Every inch of the way. At every 10th of an inch. At every 10000th of an inch. Until the intervals of rise over intervals of run become infinitesimally small.
Considering the slope at every point along a roller coaster gives an idea of what the problems are. On a roller coaster, you start at the ground level and you end up back at ground level. So if you simply use a "rise over run" rule for calculating the slope, you will get ZERO. The coaster goes up and down and up and down, but at the end, you have risen no higher than you were at the start. So the rise is zero, no matter what the run might be. So a rough application of "rise over run" to get the slope of the roller coaster route will give you ZERO.
Even though it is obvious that at (almost)every point the track is sloping up or sloping down.
So this SUGGESTS that there is more to slope than a mere application of "rise over run."
Finding the slope at any point on a roller coaster demands answers to questions such as these: what is the rise at a single point? What is the run? Can there be such a thing? Can there by a rise at a single point????? A point has no dimensions. How can it have a "rise" How can it have a "run" ?
An answer exists... It takes a little thinking about. But it can be understood. And it is at the heart of calculus.
It requires an appreciation of the fact that at any small segment of track the so called "run" can be made smaller and smaller and smaller.And the same for the rise.
Then specify a point at which the slope is to be calculated. Specify the "run" involved at that point. Specify the "rise." Do this by making approximations. A good approximation is to place the point in the middle of a small interval of distance. Calculate the slope at a point which is half way between the beginning and end points of that small interval. That seems like a pretty good way to approximate the "run." And, it is!!!
Make the interval, the run, one inch. The point then is half way between the zero inch mark and the one inch mark. Then bring the beginning and end points of that interval closer and closer to the point in question. Keeping the point half way between them. Let the point be one ten thousandth of an inch from the start of the interval; that will be a better approximation to the run at that point. Then reduce the distance to one 1 ten thousandth of an inch.. Then one millionth of an inch. Then one billionth of an inch. This leads inevitably to an interval that can be considered almost infinitesmally small.
And all the while, for these increasingly tiny intervals, there is a corresponding rise, also increasingly tiny. And all the while, as these intervals are shrinking, the ratio of rise over run is being computed, until rise over run becomes the ratio of an infinitesmally small rise over an infinitesmally small run.
And finally you have obtained the "rise over run" for a specified point. Even though a point in itself HAS NO RISE or RUN.
This serves as a plausible and persuasive process for finding the slope at ANY specified point along ANY curve. (With certain limitations t hat need not be gone into) Calculus employs techniques for computing that infinitesmal rise over infinitesimal run. That ratio is what is understood in calculus to be the "slope."
This would not be of any great practical value if not for the fact that knowing the slope at any point on many curves points the way to solving many problems of a practical nature.
The definition: "Rise over run" may not take all of this fully into account. B ecause as we have said, if we expand our interests beyond straight lines and such regular and symmetrical figures as circles, then the concept of rise over run at the dimensionless point of geometry, does not seem to have meaning. Yet, it is he essential principle at the heart of the more sophisticated calculations of calculus
So in this sense it is correct, and challenging it is not the best way of taking issue with some school of pedagogy that may put it forward.
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This was written by Peter McIntosh, a high school math teacher at Oakland Unity High School in East Oakland, California.
In his recent criticism of Khan Academy Karim Kai Ani suggested "that there's nothing revolutionary about Khan Academy at all. In fact, Khan's style of instruction is identical to what students have seen for generations." He went on to echo the concerns of many educators by criticizing Khan's approach to content delivery and pointing out flaws in his videos.
Messers Ani, Danielson and Goldenberg are obviously experts in their fields, and they make any excellent points. However, I believe that their thoughtful analysis is misdirected.
When an answer to a problem remains elusive after decades of effort by legions of passionate people, perhaps it is time to consider whether we are asking the wrong question. I believe these authors, like many American educators, are mistakenly looking for better approaches to content delivery and have missed the real problem in our math educational system, content reception.
Teachers are not failing because of ineffective content delivery; they are failing because they are not effectively addressing the character deficit in many of our students. We have spent years looking for better ways to deliver content to students who are increasingly uninterested in receiving that content.
Rather than address this root cause of educational failure, it is used as an excuse to explain the patchwork results of classroom reforms. It makes sense that the academic scores in East Oakland are lower than the scores in more affluent school districts; or that children from families where education is a priority do better than those from less focused backgrounds. However, educators need to stop using these character issues as an excuse for failed educational initiatives, and start making character their real focus.
To better understand how teachers can address this character deficit let's examine how it developed. By the time students reach the high school classroom many of them are far behind grade level in basic math concepts such as fractions. This content gap has lingered for many years. Students became accustomed to being unable to do the problems and they rationalized their constant failure. They concluded that they could not do these problems because they had a poor teacher or that they were simply not smart enough. They soon began to see each homework assignment as just more evidence confirming that they had a poor teacher or that they were not smart.
Because of these rationalizations students developed the habit of never doing homework, deepening the spiral. And this problem was compounded by the total lack of consequences. Despite not knowing the material and not doing the work, many of these students were passed on to the next grade, often with A's and B's.
These students enter the high school classroom with a content gap and a seriously skewed view of education. They have heard fractions explained dozens of times, and they have A's and B's on their transcripts. Consequently they greet much needed review lectures with false confidence. "I know this!" they say as they tune out brilliantly delivered content. Then their habit of not doing homework ensures that they do not absorb this material.
False confidence – laziness – lack of responsibility – What is this if not a character deficit? I believe we need to shift our focus from improving content delivery to helping our student repair their character. And I believe that we can do that in the classroom. But we first need to accurately name the problem.
Discussions of flaws in Khan videos are an example of misdirected focus on content delivery. The videos do have flaws, but the genius of Khan Academy is the pause button. Students have control. They can watch the videos if they choose, and they can stop any video. What is totally missed in these criticisms is the effect Khan has on student habits. And that effect is not based on the videos!
Khan Academy is an ensemble performance. Something about the design of the math exercises engages students. And the first effect is often the confrontation of the false confidence so prevalent among these students. It is amusing how many students call me over to complain that Khan has the wrong answer to a problem. But they are engaged as I explain why the Khan answer is correct.
The second part of the ensemble is the availability of "hints." For any problem students can request the detailed steps, just one to get started, or the entire solution.
The most powerful part of the ensemble may be the very natural tendency of students to help each other. They actively listen as another student explains a problem, and they become intensely focused when they are the one providing the help.
The teacher takes on a different role in a Khan classroom. The casual observer will see a teacher providing one-on-one coaching to students receptive for that guidance as they struggle on a specific problem, or providing brief explanations to small or large groups struggling with a difficult concept. The more focused observer will see a leader: defining objectives and encouraging the students to take advantage of this full ensemble of resources.
What is most interesting is the engagement. Disruptive behavior fades when the computers come out. The coaching screens and reports display amazing persistence, with students patiently working through dozens of problems until they master a topic and complete a string of correct exercises.
These students have begun to take responsibility for their education. They ask for help from peers or the teacher, or they use hints from the system. Sometimes they refer to notes to use prior problems as a model. They find a way to solve the problem in front of them and then move on to the next problem.
Oh, and some students watch videos.
Responsibility – Effort – Confidence. Real confidence based on accomplishment. A willingness to persist on difficult problems because of that confidence. Autonomy to seek help from a variety of sources. Students in a Khan classroom exhibit significant changes in their character traits. And they learn.
Interestingly, this resurrected character makes it much easier to engage the students in challenging word problems or hands-on projects, and their learning is deepened by their strengthened skills and persistence. Importantly, Khan frees teachers from more routine preparation to facilitate these deeper learning experiences.
The critics are correct on one very important point. We do need great teachers. But we need their focus to be on leadership rather than just content delivery. And we need them to use tools such as Khan Academy to reach students in ways traditional teaching cannot.
The real issue is changing the question being asked. Decades of focusing on content delivery has resulted in arguments but little success. We need to start focusing on repairing character. Khan Academy cannot replace teachers in this effort, but it provides a tool that can leverage the skill, energy and love these professionals bring to class every day.
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This was written by Ben Tilly, a computer programmer with a master's degree in math and a nearly completed doctorate, who has taught Calculus, multi-variable Calculus, linear algebra, etc.
In mathematics there are many, many cases where there are multiple equivalent ways to define things. You can choose one, and then the others are all theorems. Math doesn't care which we choose to be our definition, and therefore we should choose the simplest to understandand work with. Many high school math teachers seem to believe that whichever one they happened to be taught is the "right" one, the others are "incorrect," and that the distinction between the two has some fundamental importance. Speaking as a mathematician, absolutely nothing could be farther from the truth.
Sal Kahn used the widely used definition "rise over run" which is simple to calculate and easy to visualize. This is an excellent definition to use.
Karim Kai Ani used the definition, "slope is a rate that describes how two variables change in relation to one another." I find this definition unclear, abstract, and hard to visualize. I want students to be able to understand that slope does that, but this is NOT how I want beginning Calculus students to understand slope. The derivative is a complicated enough thing for them to understand as it is, and starting with an unclear picture of slope will just make it worse.
Both definitions can be made to work. Both appear in textbooks. I prefer Sal's.
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This was written by Evan Turner, an engineer and former instructor.
I just wanted to note that the explanation Khan is giving for slope is not particularly useful. His counter-example of calculating memory size per cost and saying it's the "inverse" slope isn't wrong, per se, but it's pointless.
From a top-level perspective, the concept of slope in elementary algebra is really just the derivative of a linear function. That is, a rate of change of a straight line. Hence, the informal definition of a derivative at a point is "the slope of the tangent line".
For example, a ball dropped and falling toward the earth will have a position of 'h - 1/2*g*t^2', where 'h' is the height (in meters) it was dropped from, 'g' is the gravitational constant (9.81 m/s^2), and 't' is how many seconds the ball has been falling. If you graph that function, where your x-axis is "time" and y-axis is "height", you get a parabola that looks like an upside-down 'U' (everything to the left of the origin can be ignored, since we don't care about the ball's velocity before we dropped it).
The derivative of position is velocity, so the slope of a line drawn tangent to any point on this parabola tells us the velocity of the ball at that instant of time. Divide the rise (change along y-axis) by the run (change in x-axis) and you get the ball's velocity. Points farther to the right, i.e. at later times, have steeper tangent lines. As the slope continues to be more and more negative, the ball is falling faster, which we know intuitively from the experience of dropping things from low and high heights.
What Khan seems to be missing, and what Karim didn't specify, is that the importance isn't that slope is change between two variables, but that it is the rate of change for a dependent variable vs. an independent one. In the case of the ball, the independent variable is time — because we can't affect its progress — and the dependent variable is position -- how far the ball has fallen after t seconds. Khan's statement about putting memory size on the y-axis and price on the x-axis is irrelevant, because the dependent variable is price and the independent variable is memory size ("how much will I pay for X gigabytes of memory?"). When you shop for an iPad, the store has signs that list the price of an iPad with 16, 32, or 64 GB of storage. This makes sense intuitively, because we want to know what we have to pay to get an iPad with the amount of storage that we want. The inverse of this would be, "I have $599, what size iPad can I afford?", by swapping the dollar and GB axes.
Regarding units, if both variables have the same unit, the units cancel out (e.g. sales tax: for X dollars worth of items purchased I pay Y dollars total, Y dollars over X dollars causes the unit of dollars to cancel out, which is why sales tax is always given as a percentage). However for velocity, we have meters fallen divided by seconds in the air, so our unit is m/s.
What is important to a student isn't just to repeat a mnemonic or formula, such as 'rise over run', but to understand what these quantities mean, why we've selected our axes a particular way, and whether the slope (rate, derivative) we've calculated is useful. In my example of the ball, it gives us a way to quantify an intuition: that dropped objects fall faster the longer it takes before they hit the ground. Telling a student "subtract y1 from y2, and divide that by x1 subtracted from x2" shows them how to get a slope, but teaches them nothing of what the slope represents.
-Evan
P.S. My background is in engineering (now a software developer), but I have spent time instructing at various levels.
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Charles McLane, a former teacher, sent emails that said in part:
It also strikes me that you are shooting the messenger in your protests about the Kahn Academy. A pathetic state of mathematics education is implied by the Kahn Academy tutorial videos: despite rather mediocre quality, the videos have achieved notable popular acclaim from struggling students, explaining to them what was not understandable in their classes. One can only speculate as to how bad those classes must be. It misses the point to criticize Kahn because he did not pay proper homage to "lesson plans" or "pedagogical intentionality...."
...and...
Slopes and rate-of-change are related, but distinct, concepts with slopes leading us gently and intuitively into the concept of rate of change. Building on both of these concepts leads us toward the beauty of calculus. Mathematics first defines its terms and then critically proceeds. It is the process of critical thought starting from whatever given definitions that defines mathematics. Kahn's definition of slope is both common and age-appropriate, if not the only possible definition. Appeal to authority for "correct" definitions is illustrative of what's wrong with math education as it totally misses the spirit of mathematics |
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American Mathematical Monthly - NOVEMBER 1998
NOVEMBER 1998
Some Characteristics of Eighth Grade Mathematics Classes in the TIMSS Videotape Study
by Alfred B. Manaster amanaster@ucsd.edu
One component of the Third International Mathematics and Science Study (TIMSS) was a videotape study of a representative sample of eighth grade mathematics classes in Germany, Japan, and the United States. This study provided a unique opportunity to examine and contrast the mathematics curriculum delivered in those classes. The author and three of his colleagues analyzed carefully constructed summary descriptions of 90 of these classes. They found aspects of the content of the lessons that provide insight into the potential of the lessons to enhance students' understanding of mathematics. This paper presents a summary of their methods and results. A representative Japanese lesson is analyzed. Special attention is given to the extent of mathematical reasoning in the lessons, the structure and coherence of the lessons, and the nature of the mathematical tasks presented during the lessons. Contrasts among countries and mathematical subjects are reported.
Summary of Conclusions About United States Lessons
·
There were no instances of explicit mathematical reasoning in the United States lessons. ·
There are more arithmetic lessons in the United States. ·
Lessons in the United States are significantly more fragmented. ·
There is less use of solver controlled and multi-step problems in the United States lessons.
The Newton and Halley Methods for Complex Roots
by Lily Yau and Adi Ben-Israel bisrael@rutcor.rutgers.edu
It is required to solve f(z) = 0 where f is analytic. Let F(x,y):=|f(x+iy)|, and let z_k=x_k+iy_k (the current iterate) be a point where f andf' are nonzero. We give a geometric interpretation of the Newton and Halley iterations for f at z_k.
Newton: The next iterate is the point closest to z_k on the intersection line of:
·
the xy-plane, and ·
the plane tangent to the graph of F at (x_k,y_k,F(x_k,y_k)).
Halley: The next iterate is the zero of the Mobius transformation that osculates the level set of |f| at z_k.
The Rook on the Half-Chessboard, or How Not to Diagonalize a Matrix
by Kiran S. Kedlaya and Lenhard L. Ng kkedlaya@math.mit.edu, lenny@math.mit.edu
We study a simple-looking family of matrices whose eigenvalues appear equally innocuous. The diagonalization of these matrices is not quite so innocuous; we present two approaches, one of which requires a four-variable combinatorial identity with a somewhat roundabout proof. We then apply this result to explore two related random walks, summarized as "the rook on the half-chessboard." Given an n-by-n chessboard from which all squares above (but not including) the northwest-southeast diagonal have been removed, a rook moves with probability 1/2 to some square in its row, and with probability 1/2 to some square in its column. We determine the speeds at which this walk, and a related, more efficient walk, become random.
The Last Round of Betting in Poker
by Jack Cassidy cassidy@sdd.hp.com
Mathematical analyses of poker usually assume an unrealistic set of circumstances that never occur in real play. Chief among these is that all players have equal hands going into the final draw.
This paper assumes generalized, unequal hand distributions for two players in the final round of betting. We develop six simple equations for optimal strategies when raising is not allowed, then look at how things change when raising is allowed.
We apply the equations to a sample hand of Seven Card Stud, and give pointers on how to apply the optimal strategies to real-life poker games.
Applications of the Universal Subjectivity of the Cantor Set
by Yoav Benyamini yoavb@tx.technion.ac.il Every compact metric space is a continuous image of the Cantor set, i.e., for each compact metric space K there is a continuous map from the Cantor set onto K.
This classical theorem of Alexandroff and Hausdorff can heuristically be rephrased as saying that "compact sets of data can be continuously encoded by the Cantor set". This principle is a very powerful tool that can be applied to solve a variety of unrelated problems in topology, geometry, and analysis. In this article we present several such applications. The following two examples are typical.
(i) (R. Grzaslewicz) For each d 1 there is a compact convex set B in R^(d+2) with the property that each compact convex subset of the d-dimensional unit cube is congruent to a face of B.
(ii) There is a real-valued, bounded, and continuous function f on the real line R with the property that for each doubly infinite sequence (y_n)_(n \ in Z) of real numbers satisfying |y_n |£ 1 for all n, there is a point t \in Rsuch that y_n = f(t + n) for all n \in Z.
NOTES
The Group Z_2 x Z_n and Regular Polygonal Paths
by Apostolos Thoma athoma@cc.uoi.gr |
Asp, Gary et al
Graphic Algebra New with no dust jacket Key Curriculum Press Paperback 4to 11" - 13" tall; 162 pages; A Key Curriculum book, quality paper, punched for three-hole notebook. This book of engaging blackline masters provides activities for algebra students to use with the graphing calculators and graphing software-technology which is rapidly becoming commonplace in the high school math classroom. Creating graphs is no longer a time consuming task for students, which leaves them more time to use graphs to study the properties of functions. Graphic Algebra helps develop new insights into algebra by providing easy-to-use lessons in which students graph and study functions using any graphing calculator or computer software for graphing.
The book helps students use graphs to solve problems set in real-world contexts; to link different representations in order to move easily between tables of values, algebraic expressions, and graphs; to develop understanding of different types of functions and their properties; to learn concepts and skills needed for graphing on a calculator or a computer; and to explore transformations of functions.
This book grew out of a research project conducted at the University of Melbourne, Australia. Graphic Algebra was designed to be used in a variety of ways to supplement and complement the teaching of algebra. Some problems can be used to introduce new ideas; others offer a novel way to review familiar ideas in a new context. The book is a perfect supplement for any curriculum involving algebra. The materials assume that students have a basic familiarity with algebraic notation and the Cartesian plane. Other prerequisite knowledge is noted for each chapter. Teachers can select short or long sequences of work designed for students at various levels. The book contains reproducible blackline masters, as well as teaching suggestions for using graphing calculators in algebra, extensive teacher notes, and appendices with specific instructions for the Texas Instruments TI-82 and TI-83, Hewlett-Packard® HP-38G, and Casio® CFX-9850G graphing calculators. For grades 8-11.; ISBN: 1559532793
Brink, Raymond W.
Intermediate Algebra Publisher: New York 1951. New with no dust jacket; Bumps on front top and bottom edges and on the corners of boards. Old price written on top of front free end paper. Light wear to boards. Otherwise very good. Appleton-Century-Crofts Second Edition Hardcover Textbook; 8vo 8" - 9" tall; 295 pages; Contains tables and answers to odd-numbered problems at the back.
Collins, S. Harold
Basic Equations New Garlic Press Paperback This series shows students in the upper grades how various mathematics disciplines can be used to solve everyday problems. 60 pages of instruction and problems to help students understand linear equations. Emphasis is placed on translating simple algebra and geometry concepts into the mathematical statements needed to solve everyday problems. Covers equations and number relationship, solving basic linear equations, word problems, and formulas. Each topic has step-by-step explanations, examples, and applications. Answers are provided. ; Applied Math Series; 60 pages; ISBN: 0921993849
Collins, S. Harold
Straight Forward: Pre-Algebra Book 1 Very Good; Lightly bent corner from shipping damage from publisher. Companion Publisher: Eugene 2007. 4to 11" - 13" tall; 63 pages; This resource teaches skills and competencies that help students make a smooth transition from basic arithmetic operations to formal algebra. Each chapter provides a step-by-step model that explains a new concept and provides examples followed by practice. A mastery test is also provided along with a glossary of basic terms. Answers are included for the exercises and mastery test at the back.; ISBN: 9781930820623
Collins, S. Harold
Straight Forward: Pre-Algebra Book 1 Book 2296
Dean, Susan
Hands On Pre-Algebra Grades 6-8 Publisher: 2007. New with no dust jacket Instructional Fair / Frank Schaffer Paperback 4to 11" - 13" tall; 80 pages; This a great supplement for grades 6-8. It is full of activities which can be adapted for varying levels of difficulty. Many activities are self-guided with questions for additional exploration. You will find a variety of pre-algebra puzzles, games and other activities which will help students to understand and enjoy math.; ISBN: 1568221843
Dyke, Frances Van
A Visual Approach to Functions Publisher: 2002. New with no dust jacket Key Curriculum Press Paperback 0.5 x 11 x 8.5 Inches; 170 pages; This series of visual exercises introduces all of the standard functions and applications students encounter in algebra. Lessons consisting of blackline masters and teacher notes and answers are organized into six chapters, each with optional activities that use graphing calculators and motion detectors. The six chapter titles are Distance as a Function of Time, Value as a Function of Time, Exponential Growth and Decay, Investments, Height of a Projectile, and Quadratic Applications.; ISBN: 9781559535373
Freeman, Sara
Algebra 2 Publisher: Dayton 2002. New with no dust jacket Milliken Publishing Company Paperback Math Reproducibles; 4to 11" - 13" tall; 48 pages; This easy-to-use workbook is sure to motivate your students and engage their interest in algebra while reinforcing the major algebra concepts. A variety of puzzles, mazes, and games will challenge students to think creatively as they sharpen their algebra skills. A special assessment section is also included to help prepare students for standardized tests. I recommend this as a reproducible supplement to an text, not as a stand-alone Algebra 2 text. Answers are provided. 48 pages. For grades 7-9; ISBN: 9780787705091
Fulton, Brad & Bill Lombard
The Pattern and Function Connection Publisher: 2001. New with no dust jacket Key Curriculum Press Paperback Blackline Activity and Homework Masters; 0.4 x 10.9 x 8.3 Inches; 161 pages; The 11 progressively paced activities, which include blackline-master worksheets and homework pages, introduce students in pre-algebra and algebra to the fundamental concept of function and its multiple representations. Students recognize, graph, analyze, and solve algebraic functions, transitioning from concrete to abstract. Full teacher notes and solutions are provided, as well as an alignment to the NCTM Principles and Standards. Punched to fit three-hole binder.; ISBN: 1559533951
Garcia, Hank
Algebra Puzzles: Build Pre-Algebra and Algebra Skills through Puzzles andProblems Publisher: Huntington Beach, CA 2006. New with no dust jacket Creative Teaching Press Paperback 4to 11" - 13" tall; 80 pages; This book uses games, puzzles, and other problem-solving activities to give students fresh, new ways of exploring learned concepts. While reviewing essential concepts and vocabulary for pre-algebra and algebra, the book helps students visualize and think more deeply about these abstract ideas. Key algebra terms are defined so that students can master the vocabulary they need. The perfect antidote to algebra anxiety.; ISBN: 9781591982333
Gray, Virginia
The Write Tool to Teach Algebra Publisher: Emeryville, CA 1993. New with no dust jacket Key Curriculum Press Paperback Textbook; 4to 11" - 13" tall; 104 pages; This activities in this book are designed to break down student resistance to mathematics with writing exercises that entertain while they promote critical thinking skills. This friendly, non-intimidating approach will encourage even your most reluctant writers to express themselves . The books contains specific class instructions, writing activities, reproducible teacher's aids, and assessment suggestions. This is a valuable resource for those who wish to address the NCTM Standards' emphasis on communication in mathematics classrooms.; ISBN: 1559530642
Hoover, Elizabeth
Tall-Tale Math: Book 2: Pre-Algebra Publisher: 2000. New Educational Impressions Paperback This series consists of Humorous Story Problems for the Middle Grades Designed to help instructors implement the current Curriculum and Evaluation Standards, these books help students understand and utilize the fundamentals of mathematics. Each book is divided into three sections: Review Sheets contain easy-to-understand definitions and examples that clearly explain conceptsSkill Builder Sheets present humorous, creative story problems; each focuses on a specific concept, which is stated at the top of the page. Extra Practice Sheets add an additional challenge for those who have mastered the previous sections. For grades 5-9: This book covers covers variables; using addition, subtraction, multiplication and division to solve equations; combining like terms; working with fractions contains parentheses and negative and positive integers; writing algebraic expressions; writing equations for word problems; and using proportion to solve equations; Tall-Tale Math; 64 pages; ISBN: 1566440572
Lovell, Robert
Probability Activities for Problem Solving & Skills Reinforcement Publisher: 1993. New with no dust jacket Key Curriculum Press Paperback 0.6 x 10.9 x 8.4 Inches; 308 pages; Punched for three-hole notebook. This book consists of a series of 61 lesson guides, each designed for one class period. The unique problems spur discussions and motivate students. They will also be reviewing their skills in whole numbers, fractions, and decimals as they improve their problem-solving skills. Palindromes, magic squares, alpha-numerics, brain busters, and problems from The Guinness Book of World Records appear in many lessons. For grades 7-11. Answers for problems are included.; ISBN: 1559530677
McCabe, John L. P.
Applying Algebra New Garlic Press Paperback This series shows students in the upper grades how various mathematics disciplines can be used to solve everyday problems. This book covers formulas; equations, ratio and proportion, and percent. Answers are provided. ; Applied Math Series; 96 pages; ISBN: 0921993636
McKell, Theresa
Algebra Puzzlers Publisher: 1998. New with no dust jacket Good Apple Paperback 4to 11" - 13" tall; 112 pages; For grades 9-12. This collection of challenging and engaging puzzles will work with students at all algebra skill levels. They can be used as warm-ups, homework, or even entire math lessons.; ISBN: 0768201012
Osterman, Wendy
Power Practice Pre-Algebra Publisher: Huntington Beach, CA 2004. New with no dust jacket Creative Teaching Press Paperback Power Practice; 4to 11" - 13" tall; 128 pages; For grades 5-8. Books in this series contain over 100 ready-to-use activity pages to provide skill practice for students. These books supplement and extend any curriculum and can be used for independent class work or homework. Answers are provided. This book correlates to the NCTM Standards. Topics include factors and multiples; decimals; rational numbers; percents; polynomials; square roots; and solving equations.; ISBN: 9781591980902
Parks, Virginia
Beginning Algebra with Applications Student Solutions Manual Publisher: Englewood Cliffs 1990. Very Good with no dust jacket; Light wear to wraps, especially at edges and corners. Fore edge has some light discoloration. Some page corner tips have been lightly bent to mark places. Pages are free of any writing. I have also listed Beginning Algebra with Applications, and I can consolidate postage when you buy them at the same time. Image used is of actual copy of book for sale, not a stock image Prentice Hall Second Printing Paperback 4to 11" - 13" tall; 228 pages; This book contains the solutions to every odd-numbered exercise and all the problems in the Chapter Tests for Beginning Algebra with Applications by Linda Exley and Vincent Smith. Punched for three-hole binder.; ISBN: 0130725889
Rasmussen, Peter (Co-Author) /king, Julie
Book 1: Operations on Integers abstract14
Rasmussen, Peter (Co-Author) /king, Julie
Book 2: Variables, Terms, and22
Rasmussen, Peter (Co-Author) /king, Julie
Book 3:30
Rasmussen, Peter (Co-Author) /king, Julie
Book 4: Polynomials49
Rasmussen, Peter (Co-Author) /king, Julie
Book 5: Rational Numbers duplicate57
Rasmussen, Peter (Co-Author) /king, Julie
Book 6: Multiplying and Divid65
Rasmussen, Peter (Co-Author) /king, Julie
Book 7: Adding and Subtract73
Rasmussen, Peter (Co-Author) /king, Julie
Book 8: Graphs using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system. For grades 5-12; Key to Algebra; 37 pages; ISBN: 1559530081
Rasmussen, Peter (Co-Author) /king, Julie
Book 9: Systems of009X
Rasmussen, Peter (Co-Author) /king, Julie
Book 10: Square Roots and Quadratic0103
Rasmussen, Steven
Answers and Notes for Algebra Books 1-4 Books 5-7 Books 8-10Algebra Reproducible Tests for Books 1-10 Publisher: 1992Saxon, John
Algebra 1: An Incremental Development Publisher: Norman, OK 1999. Good with no dust jacket; Glossy boards, no jacket, as issued. Cover is moderately worn and corners are bumped. Small black dot on bottom edge. Some writing and marks on end pages, and there are a few bits of writing in the text in pencil. Saxon Publishers, Inc. Second Edition; Eleventh Printing Hardcover Textbook; 4to 11" - 13" tall; 561 pages; Contains answers to odd-numbered problems.; ISBN: 0939798425
Saxon, John H. Jr.
Saxon Algebra I Homeschool Kit: Publisher: Norman, OK. New with no dust jacket Saxon Publishers, Inc. Third Edition Hardcover Textbook; 4to 11" - 13" tall; This kit contains everything you need to teach first year algebra at home -- a textbook, an answer key for homework, and tests with answers. This course includes five instructional components: introduction of the new increment, examples with complete solutions, practice of the increment, daily problem sets, and cumulative assessments. This course covers all topics normally introduced in first year algebra. these topics range from algebra-based real-world problems to functions and graphics -- from algebraic proofs to statistics and probability. 120 lessons. Can't determine date or number of pages without dismantling the package.; ISBN: 1565771230
Saxon, John H. Jr.
Saxon Algebra 2 Homeschool Kit: Publisher: Norman, OK 2007. New with no dust jacket Saxon Publishers, Inc. Third Edition Hardcover Textbook; 4to 11" - 13" tall; The homeschool kit contains everything you need to teach the course -- the textbook, the answer key for homework, and the tests with answers. This covers not only all the second-year algebra topics, but also a lot of geometry, including writing proof outlines. At the end of the course, your child will have completed the equivalent of one semester of informal geometry.; ISBN: 1600320163
Saxon, John H. Jr.
Saxon Homeschool Algebra 2: Lesson and Test CDs, 3rd Edition Publisher: Norman, OK. New with no dust jacket Saxon Publishers, Inc. Third Edition Non-Book Textbook; 4to 11" - 13" tall; Contains over 110 hours of Algebra 2. Contains 5 CD's; ISBN: 9781565771437
Saxon, John Jr.
Saxon Homeschool Teacher, Algebra 1 Lesson and Test CDs Publisher: Norman, OK 2008. New with no dust jacket; New book. Saxon Publishers, Inc. Second Edition CD; 4to 11" - 13" tall; This is the only official video supplement to the Saxon Math Homeschool program. It supplements, not replaces, the student textbook. Contains over 130 hours of Algebra 1.; ISBN: 1602773653
Serra, Michael
Mathercise:Classroom Warm-Up Exercises Book A (Pre-Algebra) New with no dust jacket Key Curriculum Press Paperback 4to 11" - 13" tall 7-8. This book is for middle and high school students not yet taking algebra.; ISBN: 1559530596
Serra, Michael
Mathercise: Classroom Warm-up Exercises Book B (Pre-Algebra, Algebra) New with no dust jacket Key Curriculum Press Paperback 8vo 8-9. This book is for a middle or high school student taking pre-algebra, algebra or a first-year high school math course.; ISBN: 155953060X
Shea, James
Algebra Publisher: 2001. New Steck-Vaughn Paperback These consumable workbooks provide an in-depth focus on middle school, junior high and high school concepts. Pretests and Mastery tests help place and assess students in each of these titles. Presents the basics of Algebra for students in grades 8-12. Answers are not included, but are available in a separate teacher's guide. 208 pages. New Book; Working with Numbers; 208 pages; ISBN: 0739835432
Shea, James T.
Algebra Teacher's Guide Publisher: 2002. New Steck-Vaughn Paperback This is the Answer Key and Teacher's Guide for Working with Numbers: Algebra. New Book; Working with Numbers; ISBN: 0739844113
Smith, Vincent K. and Exley, Linda L.
Beginning Algebra with Applications Publisher: Englewood Cliffs 1994. Very Good with no dust jacket; Pages are clean and have no writing. Corners are bumped and one of them is torn. There is some chipping on the top front edge. Spine has a wrinkle near the top. Price sticker on back Prentice Hall Second Edition; First Printing Hardcover Textbook; 4to 11" - 13" tall; 568 pages; ISBN: 0130672572
Winter, Mary Jean & Ronald J. Carlson
Probability Simulations with CD-Rom Publisher: 2000. New with no dust jacket Key Curriculum Press Paperback 0.47 x 10.87 x 8.43 Inches; 144 pages; The investigations in this book reinforce the basic concepts of probability, can be used to enrich any mathematics curriculum, and are ideal for extended class periods. Any graphing calculator with a random-number generator can be used to complete the 13 probability simulations. Calculator programs for the activities can be downloaded from the accompanying CD onto Texas Instruments TI-82 and TI-83, and Casio™ CFX-9970, 9870, and 9850 graphing calculators, entered by hand from the program listings, or adapted to other calculators following the program annotations. This book will supplement courses in Algebra 1, Algebra 2, integrated algebra, and precalculus. Book is punched to fit into three-hole binder. Answers are provided.; ISBN: 155953401X |
uCalc Graphing Calculator 4.6 description
uCalc Graphing Calculator 4.6 is released as a functional and fantastic math class companion for high school and college students which is equally useful to anyone who needs a multi-purpose calculator. It includes an expression evaluator, where you can enter your calculations; a unit converter; an equation solver; user solution modules; an adder and more. It can also plot equations (both 2D and 3D). It comes with an animated tutorial that shows the calculator's capabilities. The user can program the animated character as well, and get it to speak, gesture, and move around.
Multi-purpose calculator includes: Equation plotter; Expression Evaluator; Many built-in functions and operators; Programmable cartoon character (useful for tutorials); User Solution Modules (BMI, Percentage Increase, various financial, create your own); Date/time constructs (enter "15 years ago" or "48 hours from now" to see what date it was or will be); Unit converter; Define your own functions and variables; uCalc is very programmable.
Major Features:
Interactive Animated Tutorial -- Sit back and watch (and interact) as an animated character walks you through many of the calculator's features. You may likewise create your own tutorials to demonstrate math lessons by programming the animated character.
Unit Converter -- You can convert between units of measure in a flexible manner. You'll find categories such as Length, Capacity, Weight, Temperature, Time, and more. You can even add your own categories and units.
User Solution Modules -- You can interactively define and solve problems by simply filling in the blanks, without directly having to manipulate math formulas. To get you started, uCalc comes with useful financial modules that will let you calculate things like add-on interest, compound interest, ordinary annuity sum, present value (mortgage), and more. You might find yourself eventually creating your own modules to solve all kinds of other problems with this unique feature.
Graphing -- You can plot Cartesian, polar, parametric, 3D equations, or coordinates from data files. You can print your graphs, or save them for later use.
Equation Solver -- You can solve for the unknown value in an equation. If your equation has several solutions, this calculator can display them all. If there are several solutions, but you are interested in only the negative ones for instance, you can enter the solution range you want uCalc to work in.
User Functions & Variables -- You can define variables with meaningful names, and functions with multiple parameters for use anywhere in the program.
Summation Tables -- You can use this general mathematical tool for practical things such as calculating annuity, compound growth, etc...
Integrator -- You can do numerical integrations as well.
Saving/Loading -- You can save your work, and load it later, or paste it to the clipboard in various convenient ways.
Adder -- If you want to add up a bunch of numbers, such as prices alongside the product descriptions in your shopping list, you can do this conveniently with the Adder.
Enhancements:
Version 4.6 includes additional date/time constructs and improves equation solver and plotting.
Free and easy biorhythm calculator. Knowing your biorhythms, you can be sure about what to expect from each day in order to succeed at business, or in sports, exams or job interviews, negotiating or d Free Download
A new version of Label-Creator. The best Freeware Label printing program. Print your own cd label. It is easy to create CD! This program enables you to print your own CD jacketsfor a complete package! Free Download |
Course Descriptions
I. Primarily for Undergraduates
A broad overview
of the subject, intended primarily for liberal arts students.
What do mathematicians do, and why do they do it? We will examine
the art of proving theorems, from both the philosophical and
aesthetic points of view, using examples such as non-Euclidean
geometries, prime numbers, abstract groups, and uncountable sets.
Emphasis will be placed on appreciating the beauty and variety
of mathematical ideas. The course will include a survey of important
results and unsolved problems that motivate mathematical research.
An
interdisciplinary exploration into the creative process and use
of imagination as they arise in the study of mathematics and
poetry. The goal of the course is to guide each participant towards
the experience of independent discovery, be if of a new insight
into a math problem or an overlooked aspect of a poem. Students
with and without backgrounds in either subject are welcome --
no calculus will be required. No prerequisites. Enrollment limited
to 35. Written permission required.
0040
Calculus
and Its History (History of Mathematics 4)
In this course,
students interested in learning why the calculus is justly described
as one of the greatest achievements of the human spirit will
find its concepts and techniques made more accessible by being
placed in historical context. Beginning with the roots of calculus
n the classical mathematics of antiquity, we will trace its development
through the Middle Ages to the work of Newton and Leibniz and
beyond. At each stage, we will examine the philosophical and
practical challenges to existing mathematics that spurred this
continuing development. While the course is aimed primarily at
non science concentrators, it will also provide a thorough exposition
of the basic techniques of calculus useful for further study
of science and mathematics.
A
slower-paced introduction to calculus for students who require
additional preparation for calculus. This sequence presents the
same calculus topics as Mathematics 9, together with all the
necessary pre-calculus topics. Students successfully completing
this sequence will be prepared for Mathematics 10. Placement
in this course requires permission of the instructor.
A one-semester
introduction to calculus recommended for students who wish to
learn the basics of calculus for application to social sciences
or for cultural appreciation as part of a broader education.
Topics include functions, equations, graphs, exponentials and
logarithms, and differentiation and integration; applications
such as marginal analysis, growth and decay, optimization, and
elementary differential equations. May not be taken for credit
in addition to MA 9
The
course treats topics in geometry of four and higher dimensions,
related to different parts of mathematics as well as interrelations
with physical and biological sciences, literature, cognitive
science, philosophy, and art. There are substantial writing assignments
and final projects, involving mathematical and non-mathematical
topics. There are no prerequisites.
An intensive course
in the calculus of one variable including limits; differentiation;
maxima and minima, and the chain rule for polynomials, rational
functions, trigonometric functions, and exponential functions.
Introduction of integration with applications to area and volumes
of revolution. Mathematics 9 and 10 or the equivalent are recommended
for all students intending to concentrate in mathematics or the
sciences. May not be taken in addition to 5, 6, or 7; 10 may
not be taken in addition to MA 17.
A
continuation of the material of Mathematics 9 including further
development of integration, inverse trigonometric and logarithmic
functions, techniques of integrations, and applications which
include work and pressure. Other topics covered are infinite
series, power series, Taylor's formula, polar coordinates, parametric
equations, introduction to differential equations, and numerical
methods. Mathematics 9 and 10 or the equivalent are recommended
for all students intending to concentrate in mathematics or the
sciences.
This course begins
with a review of fundamentals of calculus, and includes infinite
series, power series, paths, and differential equations of first
and second order. Placement in this course is determined by the
department on the basis of high school AP examinations scores
or the results of tests given by the department during orientation
week. May not be taken in addition to MA 10.
Three-dimensional
analytic geometry. Differential and integral calculus of functions
of two or three variables: partial derivatives, multiple integrals,
Green's Theorem. Prerequisite: Mathematics 10 or 17. Mathematics
18 may not be taken in addition to MA 20 or MA 35.
This course, which
covers roughly the same material and has the same prerequisites
as Mathematics 17, is intended for students with a special interest
in physics or engineering. The main topics are: calculus of vectors
and paths in two and three dimensions; differential equations
of the first and second order; and infinite series, including
power series and Fourier series.
This
course, which covers roughly the same material as Mathematics
18, is intended for students with a special interest in physics
or engineering. The main topics are: geometry of three-dimensional
space; partial derivatives; Lagrange multipliers; double, surface,
and triple integrals; vector analysis; Stokes' theorem and the
divergence theorem, with applications to electrostatics and fluid
flow. Prerequisite: MA 10, MA 17, or MA 19.
This
course will provide an overview of one of the most beautiful
areas of mathematics. It is ideal for any student who wants a
taste of mathematics outside of, or in addition to, the calculus
sequence. Topics to be covered include: prime numbers, congruences,
quadratic reciprocity, sums of squares, Diophantine equations,
and as time permits, such topics as cryptography and continued
fractions. No prerequisites.
Linear
algebra for students of greater aptitude and motivation. Recommended
for prospective mathematics concentrators, and science and engineering
students who have a good mathematical preparation. Topics include:
matrices, linear equations, determinants, characteristic polynomials,
and eigenvalues; vector spaces and linear transformations; inner
products; Hermitian, orthogonal, and unitary matrices; bilinear
forms; elementary divisors and Jordan normal forms. Provides
a deeper and more extensive treatment of the topics in MA 52
and can be substituted for MA 52 in fulfilling requirements.
Prerequisite: MA 18, MA 20, or MA 35.
II. For Undergraduates and
Graduates
Completeness properties
of the real number system, topology of the real line. Proof of
basic theorems in calculus, infinite series. Topics selected
from ordinary differential equations. Fourier series, Gamma functions,
and the topology of Euclidean plane an 3D-space. Prerequisite:
MA 18, MA 20, or MA 35. MA 52 or MA 54 may be taken concurrently.
Most students are advised to take MA 101 before MA 113.
The
topics covered are chosen from Euclidean, non-Euclidean, projective,
and affine geometry. This course is highly recommended for all
students who are considering teaching high school mathematics.
Prerequisite: MA 52, MA 54, or permission of the instructor.
The study of curves
and surfaces in 2- and 3-dimensional Euclidean space using the
techniques of differential and integral calculus and linear algebra.
Topics include curvature and torsion of curves, Frenet-Serret
frames, global properties of closed curves, intrinsic and extrinsic
properties of surface, Gaussian curvature and mean curvatures,
geodesics, minimal surfaces, and the Gauss-Bonnet theorem.
Ordinary
differential equations including existence and uniqueness theorems
and the theory of linear systems. Topics may also include stability
theory, the study of singularities, and boundary value problems.
The wave equation,
the heat equation, Laplace's equation, and other classical equations
of mathematical physics and their generalizations. Solutions
in series of eigenfunctions, maximum principles, the method of
characteristics, Green's functions, and discussion of well-posedness
problems.
Calculus
on manifolds. Differential forms, integration, Stokes' formula
on manifolds, with applications to geometrical and physical problems,
the topology of Euclidean spaces, compactness, connectivity,
convexity, differentiability, and Lebesgue integration. It is
recommended that a student take a 100-level course in analysis
before attempting MA 113.
Infinite-dimensional
vector spaces, with applications to some or all of the following
topics: Fourier series and integrals, distributions, differential
equations, integral equations, and calculus of variations. Prerequisite:
at least one 100-level course in Mathematics or Applied Mathematics
or permission of the instructor.
An
introduction to the principles and concepts of modern abstract
algebra. Topics will include groups, rings, and fields, with
applications to number theory, the theory of equations, and geometry.
MA 153 is required of all students concentrating in mathematics.
Galois theory together
with selected topics in algebra. Examples of subjects which have
been presented in the past include algebraic curves, group representations,
and the advanced theory of equations. Prerequisite: MA 153. May
be repeated for credit.
Topics include symmetric
ciphers, public key ciphers, complexity, digital signatures,
applications and protocols. Math 153 will not be required for
the course. What is needed from abstract algebra and elementary
number theory will be covered.
Prerequisite: MA 52 or 54.
Topics
in special areas of mathematics not included in the regular course
offerings. Offered from time to time when there is sufficient
interest among qualified students. Contents and prerequisites
vary. Written permission required.
Introduction to the
theory of analytic functions of one complex variable. Content
varies somewhat from year to year, but always includes the study
of power series, complex line integrals, analytic continuation,
conformal mapping, and an introduction to Riemann surfaces.
The
theory of the classical partial differential equations as well
as the method of characteristics and general first order theory.
Basic analytic tools include the Fourier transform, the theory
of distributions, Sobolev spaces, and techniques of harmonic
and functional analysis. More general linear and nonlinear elliptic,
hyperbolic, and parabolic equations and properties of their solutions,
with examples drawn from physics, differential geometry, and
the applied sciences. Generally, semester II of this course concentrates
in depth on several special topics chosen by the instructor.
Introduction to algebraic
and analytic number theory. Topics covered during the first semester
include number fields, rings of integers, primes and ramification
theory, completions, adeles and ideles, and zeta functions. Content
of the second semester will vary from year to year; possible
topics include class field theory, arithmetic geometry, analytic
number theory, and arithmetic K-theory. Prerequisite: MA 251.
This
course introduces probability spaces, random variables, expectation
values, and conditional expectations. It develops the basic tools
of probability theory, such fundamental results as the weak and
strong laws of large numbers, and the central limit theorem.
It continues with a study of stochastic processes, such as Maarkov
chains, branching processes, martingales, Brownian motion, and
stochastic integrals. Students without a previous course in measure
and integration should take MA 221 (or Applied Math 211) concurrently. |
Skills
All the skills below should be trained by solving problems, writing down their solutions, evaluating the solutions with the help of TA/teacher or by comparing to available solutions, and reflecting on the weaknesses identified in the evaluation.
Meta-skills:
Understand what a given problem asks for
Identify strategies and techniques that might help with solving the problem
Formulate a strategy that will lead to the solution of the problem
Follow this strategy performing necessary calculations along the way
Adjust the strategy based on new information discovered while trying to implement it
Write up the obtained solution in a way that makes it readable
Critically evaluate the written solution
Skills:
Find the equation of a line given two points on the line/point on a line + its slope
Given an equation of a line identify its slope
Find the slope of a line parallel/perpendicular to a given one
Find the equation of a circle given its radius and center
Given an equation of a circle identify its radius and center
Identify a given equation as the equation of a line or a circle. Sketch the corresponding geometrical shape.
Find points of intersection of two lines/a line and a circle/two circles
Solve a quadratic equation
Find the distance between two points
Find the length of the third side in a right-angled triangle with given lengths of the other two sides.
Find the lengths of all the sides and magnitudes of all the angles in a right-angled triangle with given length of one of the sides and given magnitude of one of the angles that is not the right angle.
Given a function f having a relatively simple algebraic expression that involves sums/differences, powers of x, quotients, products, trigonometric functions, logarithms, exponentiation and possibly a definition by cases,
Find its domain and range
Sketch its graph (for very simple functions)
Find where it is continuous
Find where it is differentiable
Find an algebraic expression for the composition of two or more functions
Find the average rate of change/slope of the secant line of a function on an interval
Find the instantaneous rate of change/slope of the tangent line of a function at a point
Find the derivatives of simple functions using the definition of the derivative
Identify instantaneous velocity as the derivative of position with respect to time and instantaneous acceleration as the derivative of the velocity with respect to time
Evaluate (possibly one-sided) limits of functions
Sketch/find an algebraic expression for a function that satisfies some given properties
Show that there is at least one/several solutions of a given equation on a given interval using Intermediate Value Theorem or using some algebraic technique/guessing a solution.
Find slope of the tangent line to a curve given by an equation of the form f(x,y)=g(x,y) at a point lying on the curve.
Simplify expressions that involve exponents and logarithms
Evaluate trigonometric functions at some values of the argument
Solve simple trigonometric equations (e.g. sin x = 1/2)
Work with angles measured in radians
Apply the chain rule in situations where the expression to be differentiated involves unknown functions
Know when and how to apply an exponential growth model
Determine whether a statement is true or false. Prove it if it is true or find a counterexample if it is false (the statement is some statement about lines, circles, functions, limits, continuity, differentiability, derivatives, existence of solutions to some equations, differentiation rules, trigonometric functions, exponentiation, logarithms or curves defined by an equation). |
Students' understanding and application of the area under the curve concept in physics problems
Phys. Rev. ST Phys. Educ. Res. 7, 010112
–
Published 28 June 2011
Dong-Hai Nguyen and N. Sanjay Rebello
Abstract
This study investigates how students understand and apply the area under the curve concept and the integral-area relation in solving introductory physics problems. We interviewed 20 students in the first semester and 15 students from the same cohort in the second semester of a calculus-based physics course sequence on several problems involving the area under the curve concept. We found that only a few students could recognize that the concept of area under the curve was applicable in physics problems. Even when students could invoke the area under the curve concept, they did not necessarily understand the relationship between the process of accumulation and the area under a curve, so they failed to apply it to novel situations. We also found that when presented with several graphs, students had difficulty in selecting the graph such that the area under the graph corresponded to a given integral, although all of them could state that "the integral equaled the area under the curve." The findings in this study are consistent with those in previous mathematics education research and research in physics education on students' use of the area under the curve.
DOI:
Received 30 January 2011
Published 28 June 2011
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Students
APS Members
ISSN 1554-9178 (online). Use of the American Physical Society websites and journals implies that
the user has read and agrees to our Terms and
Conditions and any applicable
Subscription
Agreement.
Physical Review Special Topics - Physics Education Research ® is a trademark of the American Physical Society. |
Geometry - Second Edition is a clear presentation of the essentials of geometry for the high school student. Topics include: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transformations. Volume 2 includes the last 6 chapters: Similarity, Right Triangle Trigonometry, Circles, Perimeter and Area, Surface Area and Volume, and Rigid Holy Grail of mathematics revealed as a truly 17th-century numerical and geometrical proof as a letter by Fermat to a colleague. This will withstand all challenges 2 includes the last 7 chapters and covers the following topics: Sampling Distributions and Estimations, Hypothesis Testing, Regression and Correlation, Chi-Square, Analysis of Variance and F-Distribution, and Non-Parametric Statistics. It also includes a collection of resources in the final chapter |
Hate her so much. She doesn't know how to clarify steps clearly when explaining things. Gives up too easily when she can't explain something right. My cat can teach me algebra better than she can. If you have her, don't even listen to her lecture. You'll have an easier time learning it out of the book.
After reading the rules above, please tell us why you are flagging this rating. |
Cengage Advantage Books: Developmental Mathematics ADVANTAGE SERIES of C. L. Johnston, Alden T. Willis, and Gale M. Hughes' DEVELOPMENTAL MATHEMATICS is a traditional, straight-forward, extremely popular book which is noted for its one-step, one-concept-at-a-time approach. All major topics are divided into small sections, each with its own examples and often with its own exercises--an approach that helps students master each section before proceeding to the next one. As part of the ADVANTAGE SERIES, this version will offer all the quality content you've come to expect from Johnston, Willis, and Hughes sold to your students at a significantly lower price. |
Find a series of manipulations using the basic iden tities to sho w that ... W orksheet for graphing trig functions 1. Let y = 2 cos (4 x ) What is the p erio d of this ...
37 Trigonometry According to the Mathematics: Content ... skills in five topic areas: The six basic trigonometric functions ... You can use your graphing calculator to solve ...
Graph of the Trigonometric Functions Dr. Philippe B. Lav al Kennesaw State University April 17,2005 Abstract This handout discusses the graph of the six trigonometric ...
Functions, Statistics and Trigonometry Welcome to FST! Students taking FST should have good basic skills and successfully ... will be a worksheet or worksheets ... procedures 2008 fst.pdf
... preparing students touse trigonometry as it ... Graphing and Inverse Functions 103 5.1 Graphing the ... Before discussing those functions, we will review some basic ...
... waves is instrumental to the student of trigonometry. This lesson presents the basicgraphing ... of constants on generalized sine and cosine functions. When graphing ...
A basic skills item can be solved by performing a sequence of ... the Trigonometry Placement Test include: Trigonometric functions and identities Right-triangle trigonometry ... |
The author's. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many...
Praise for the First Edition
"...a reference for everyone who is interested in knowing and handling uncertainty."
—Journal of Applied Statistics
The critically acclaimed First Edition of Understanding Uncertainty provided a study of uncertainty addressed to scholars in all fields, showing that uncertainty could be measured by probability, and that probability obeyed three basic rules that enabled uncertainty to be handled sensibly in everyday life. These ideas were extended to embrace the scientific method and to show how decisions, containing an uncertain element, could be...
A comprehensive account of the theory and application of Monte Carlo methods
Based on years of research in efficient Monte Carlo methods for estimation of rare-event probabilities, counting problems, and combinatorial optimization, Fast Sequential Monte Carlo Methods for Counting and Optimization is a complete illustration of fast sequential Monte Carlo techniques. The book provides an accessible overview of current work in the field of Monte Carlo methods, specifically sequential Monte Carlo techniques, for solving abstract counting and optimization problems.
Written by authorities in...
Moduli theory is the study of how objects, typically in algebraic geometry but sometimes in other areas of mathematics, vary in families and is fundamental to an understanding of the objects themselves. First formalised in the 1960s, it represents a significant topic of modern mathematical research with strong connections to many areas of mathematics (including geometry, topology and number theory) and other disciplines such as theoretical physics. This book, which arose from a programme at the Isaac Newton Institute in Cambridge, is an ideal way for graduate students and more experienced...
Focusing on what actuaries need in practice, this introductory account provides readers with essential tools for handling complex problems and explains how simulation models can be created, used and re-used (with modifications) in related situations. The book begins by outlining the basic tools of modelling and simulation, including a discussion of the Monte Carlo method and its use. Part II deals with general insurance and Part III with life insurance and financial risk. Algorithms that can be implemented on any programming platform are spread throughout and a program library written in R...
Interest in the skew-normal and related families of distributions has grown enormously over recent years, as theory has advanced, challenges of data have grown, and computational tools have made substantial progress. This comprehensive treatment, blending theory and practice, will be the standard resource for statisticians and applied researchers. Assuming only basic knowledge of (non-measure-theoretic) probability and statistical inference, the book is accessible to the wide range of researchers who use statistical modelling techniques. Guiding readers through the main concepts and, as well as philosophy, where judgment aggregation itself originates from, but have recently captured the attention of disciplines like computer science, artificial intelligence and multi-agent systems. Judgment aggregation has emerged in the last...
Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical foundation for convex optimization, having deep knowledge of convex analysis helps students and researchers apply its tools more effectively. The main goal of this book is to provide an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern techniques of variational analysis are employed to...
THE COMPLETE COLLECTION NECESSARY FOR A CONCRETE UNDERSTANDING OF PROBABILITY
Written in a clear, accessible, and comprehensive manner, the Handbook of Probability presents the fundamentals of probability with an emphasis on the balance of theory, application, and methodology. Utilizing basic examples throughout, the handbook expertly transitions between concepts and practice to allow readers an inclusive introduction to the field of probability.
The book provides a useful format with self-contained chapters, allowing the reader easy and quick reference. Each chapter includes an...
Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this book is to present a survey of work on the computational aspects of cooperative game theory. We begin by formally defining transferable utility games in characteristic function form, and introducing key solution concepts such as the core and the Shapley value. We then discuss two major issues that arise when considering such games from a computational perspective: identifying compact...
Differential forms are utilized as a mathematical technique to help students, researchers, and engineers analyze and interpret problems where abstract spaces and structures are concerned, and when questions of shape, size, and relative positions are involved. Differential Forms has gained high recognition in the mathematical and scientific community as a powerful computational tool in solving research problems and simplifying very abstract problems through mathematical analysis on a computer. Differential Forms, 2 nd Edition, is a solid resource for students and professionals...
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion applications. Swarm-based software and source codes are available on his website such as completeness, compactness and... |
Southeastern Algebra. I use it for budgets, graphing, printing labels among other things. |
9780321577788 is an easy-to-follow, step-by-step guide on how to use the TI-83/84 Plus and TI-89 graphing calculators that follows the sequence of topics in the text. It provides worked-out examples to help students fully understand and use the graphing calculator. |
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Using interactive manipulations created with Mathematica, the Illinois State Water Survey studies groundwater recharge and discharge, as deemed critical by the National Research Council. Hydrogeologist Yu-Feng Lin explains the advantages gained from using Mathematica in this video.
In this video, get a quick introduction to the Wolfram Education Portal, which features teaching and learning tools created with Mathematica and Wolfram|Alpha, including a dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more Chinese Portuguese Russian Spanish audio Includes Spanish audio.
Paul Abbott, a faculty member in the school of physics at the University of Western Australia, uses Mathematica to build courseware, from lectures to exams. His students visualize surfaces, explore concepts interactively, hypothesize results, and check their work—all in Mathematica.
This screencast provides an overview of functionality in Mathematica 7 that makes it easy for educators to integrate into precollege, community college, and higher-education classrooms. It includes many examples of Mathematica's use for education that can be implemented immediately.
In this Wolfram Mathematica Virtual Conference 2011 course, learn why Mathematica is used for academic research with a look at its programming language, support for parallel computing, and multiple publishing and deployment options.
In this Wolfram Mathematica Virtual Conference 2011 course, learn different ways to use Mathematica to enhance your calculus class, such as using interactive models and connecting calculus to the real world with built-in datasets.
computerbasedmath.org is a project to build a new math curriculum with computer-based computation at its heart. In this talk from the Wolfram Technology Conference 2011, Conrad Wolfram discusses the concept, progress, and plans.
Mathematica can be used to enhance course management systems by helping teachers easily communicate ideas, give students immediate feedback, and link real-world datasets to textbook examples. Learn more in this screencast. |
Math Study Skills-Workbook - 4th edition
Summary: This workbook helps learners identify their strengths, weaknesses, and personal learning styles--and then presents an easy-to-follow system to increase their success in mathematics. With helpful study tips and test-taking strategies, this workbook can help reduce ''math anxiety'' and help readers become more effective at studying and learning mathematics 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase benefits world literacy!90 +$3.99 s/h
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OBJECTIVES: To provide the
fundamental analytical and numerical tools for analysis and modeling of
vibration phenomena in discrete and continuum SDOF and MDOF linear systems. Learning of advanced analytical tools and methods for experimental
identification of system parameters using recorded data, i.e., frequency domain
parameter identification methods.
Read/understand and practice: The lecturer when grading
your work will make an effort to READ
your solution to problems in an exam or homework. Correct usage of English
language with explanations of procedures using full sentences will make a large portion of your partial grade.
Worked problems that show a jargon of numbers without definitions and
procedures will MERIT a LOW grade, even if the (numerical) answer should be
correct.
Recommended problems from your textbook (Be
ahead of the game. These problems may be part of your graded homework)
week
chapter
Problem numbers
1
1
5, 8, 13, 14, 15, 20, 43, 44, 56
2
2
3&4
3
3, 7, 19, 20, 22, 25, 50, 52
5
1
Derive EOMs MODF systems: 25, 27, 30, 33, 36, 38, 39, 44, 49
6
4
10,19, 21, 30, 36, 39,34, 55
7
5
14, 18, 29
8
7
3, 11, 43, 49
10
6
3, 9, 11, 14, 15, 28, 38, 54
11-14
More to follow
Class Notes (handouts
& worked examples)
recommendation: Download handouts as
needed. The lecturer may update the notes as the semester progresses. Check syllabus to realize which handout(s) will be
used in a given week.
warning:Pps (power point slide shows) may not work on your Mac,
Tablet, etc.
Introduction
to motion in mechanical systems. Definition of design, analysis, and testing.
Steps in Modeling. Continuous and lumped parameter systems. Second Order
Systems and differential equations of motion. Definitions of Free and Forced
Responses. The purpose of analysis and relevant issues to resolve.
The handouts and textbook
used in this course are copyrighted. By "handouts," I mean all
materials generated for this class, which include but are not limited to
syllabi, quizzes, exams, lab problems, in-class materials, review sheets, and
additional problem sets. Because these materials are copyrighted, you do not
have the right to distribute freely the handouts, unless the author expressly
grants permission. |
Learn about linear graphs and vectors simply by just a swipe. Select graphs or vectors on intro page.
Find the relationship between lines, gradients, vectors and equations by drawing a graph with your finger while this app does the rest.
No input of values is necessary. This is an automatic graph app.
Math Graph app – you draw graph, it works out the equation for you.
Trace the graph with your finger. When done, sit back and study your graph.
The app will draw the line and calculate gradient.
It will also write down the graph equation at bottom left of screen.
On vectors screen, you draw vectors with your finger, and this app analyzes them for you. It is an easy way to learn about simple vectors including addition of vectors.
This native math app is meant for beginners in graphical methods, the first 2 years of learning graphs.
If you have studied complex graphical methods but would still like to remind yourself about the basics, this app will do just that.
You can work with the equations produced at your leisure, re-arranging them to discover more about graphical maths58 total
5 30
4 5
3 3
2 2
1 18
A Google User
good graph tool for basic math This app is a very good tool to study simple graphs and to analyze equations of a line. I like using it when relaxing in bedA Google UserSimilarGraph 89 - Emulator for the TI-89 graphing calculator. The ROM image is not included. You need to extract this from you own calculator. Make sure you have the ROM ready before downloading this emulator.
The program allows draw functions graphs in specified area. The area is defined by the X min, X max, Y min, Y max values and division values for X and for Y. Functions are defined by the expression string and color. You should click on function string in the table to edit it. Only "x" symbol should be used as a varible. After area and functions data input click "Redraw" button to draw it. On phones graph is drawn on a full screen. Via the menu You can maximize graph on the full screen, get help about functions, about program, save the graph as a picture or sent e-mail with it (with help of mail client app,such as gmail). The advertisement will be shown after the 5 redraws, and may be closed via the menu (or shown). And please, sometimes click advertisement - it may help for developer))). |
More About
This Textbook
Overview
This fifth edition of Precalculus Mathematics has been written to provide the essential concepts and skills of algebra, trigonometry, and the study of functions that are needed for further study in mathematics |
Algebra textbook is a college-level, introductory textbook that covers the important subject of Algebra -- one of the basic building blocks of studies in higher mathematics. Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.This textbook covers:The Building Blocks of Algebra -- Real Numbers, Exponents, Scientific Notation, Order of Operations, Working with Polynomials, Factoring, Rational Expressions, Radical Notation and Exponents, Basics of Equation SolvingGraphs, Functions, and Models -- Graphing, Functions: An Introduction, Modeling Equations of Lines, Functions Revisited, Algebra of Functions, TransformationsFunctions, Equations, and Inequalities -- Linear Equations and Functions, Complex Numbers, Quadratic Equations, Functions, and Applications, Graphs of Quadratic Functions, Further Equation Solving, Working with Linear InequalitiesPolynomial and Rational Functions -- Polynomial Functions and Models, Graphing Polynomial Functions, Polynomial Division; The Remainder and Factor Theorems, Zeroes of Polynomial Functions and Their Theorems, Rational Functions, Inequalities, Variation and Problem SolvingExponents and Logarithms -- Inverse Functions, Graphing Exponential Functions, Graphing Logarithmic Functions, Properties of Logarithmic Functions, Growth and Decay; Compound InterestSystems of Equations and Matrices -- Systems of Equations in Two Variables, Systems of Equations in Three Variables, Matrices, Matrix Operations, Inverses of Matrices, Determinants and Cramer's Rule, Systems of Inequalities and Linear Programming, Partial FractionsConic Sections -- The Parabola, The Circle and the Ellipse, The Hyperbola, Nonlinear Systems of Equations and InequalitiesSequences, Series and Combinatorics -- Sequences and Series, Arithmetic Sequences and Series, Geometric Sequences and Series, Mathematical Inductions, Combinatorics, The Binomial Theorem, Probability''This free online textbook (e-book in webspeak) is a one semester course in basic analysis. This book started its life as my...
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'This free online textbook (e-book in webspeak) is a one semester course in basic analysis. This book started its life as my lecture notes for Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009, and was later enhanced to teach Math 521 at University of Wisconsin-Madison (UW-Madison). A prerequisite for the course is a basic proof course. It should be possible to use the book for both a basic course for students who do not necessarily wish to go to graduate school, but also as a first semester of a more advanced course that also covers topics such as metric spaces.'
'Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data...
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'Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data related to real life sciences problems. Second, the ultimate goal of calculus in the life sciences primarily involves modeling living systems with difference and differential equations. Understanding the concepts of derivative and integral are crucial, but the ability to compute a large array of derivatives and integrals is of secondary importance. Third, the depth of calculus for life sciences students should be comparable to that of the traditional physics and engineering calculus course; else life sciences students will be short changed and their faculty will advise them to take the 'best' (engineering) course.In our text, mathematical modeling and difference and differential equations lead, closely follow, and extend the elements of calculus. Chapter one introduces mathematical modeling in which students write descriptions of some observed processes and from these descriptions derive first order linear difference equations whose solutions can be compared with the observed data. In chapters in which the derivatives of algebraic, exponential, or trigonometric functions are defined, biologically motivated differential equations and their solutions are included. The chapter on partial derivatives includes a section on the diffusion partial differential equation. There are two chapters on non-linear difference equations and on systems of two difference equations and two chapters on differential equations and on systems of differential equation.'
'This book is designed for the transition course between calculus and differential equations and the upper division...
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'This book is designed for the transition course between calculus and differential equations and the upper division mathematics courses with an emphasis on proof and abstraction. The book has been used by the author and several other faculty at Southern Connecticut State University. There are nine chapters and more than enough material for a semester course. Student reviews are favorable.It is written in an informal, conversational style with a large number of interesting examples and exercises, so that a student learns to write proofs while working on engaging problems.'
'Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum...
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'Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the pro-cesses of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are as follows:To help students learn how to read and understand mathematical definitions and proofs;To help students learn how to construct mathematical proofs;To help students learn how to write mathematical proofs according to ac-cepted guidelines so that their work and reasoning can be understood by others; andTo provide students with material that will be needed for their further study of mathematics.'
'This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially but not...
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'This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially but not exclusively on the part of combinatorics that mathematicians refer to as "counting." The book consists almost entirely of problems. Some of the problems are designed to lead you to think about a concept, others are designed to help you figure out a concept and state a theorem about it, while still others ask you to prove the theorem. Other problems give you a chance to use a theorem you have proved. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works. There are problems that some people will solve quickly, and there are problems that will take days of thought for everyone. Probably the best way to use this book is to work on a problem until you feel you are not making progress and then go on to the next one. Think about the problem you couldn't get as you do other things. The next chance you get, discuss the problem you are stymied on with other members of the class. Often you will all feel you've hit dead ends, but when you begin comparing notes and listening carefully to each other, you will see more than one approach to the problem and be able to make some progress.In fact, after comparing notes you may realize that there is more than one way to interpret the problem. In this case your first step should be to think together about what the problem is actually asking you to do. You may have learned in school that for every problem you are given, there is a method that has already been taught to you, and you are supposed to figure out which method applies and apply it. That is not the case here. Based on some simplified examples, you will discover the method for yourself. Later on, you may recognize a pattern that suggests you should try to use this method again.' |
Intermediate Algebra carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics,...
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Intermediate Algebra carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics, enhancing it all with with the modern amenities that only a free online text can deliver.It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines. Traditionally, the study of algebra is separated into a two parts, Elementary and Intermediate Algebra. This textbook by John Redden, Intermediate Algebra, is the second part. Written in a clear and concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study in applications found in most disciplines. Used as a standalone textbook, Intermediate Algebra offers plenty of review as well as something new to engage the student in each chapter. Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra. While CAS independent, a standard scientific calculator will be required and further research using technology is encouraged. Intermediate Algebra is written from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success. A more modernized element, embedded video examples, are present, but the importance of practice with pencil and paper is consistently stressed. Therefore, this text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today. The importance of Algebra cannot be overstated; it is the basis for all mathematical modeling used in all disciplines. After completing a course sequence based on Elementary and Intermediate Algebra, students will be on firm footing for success in higher-level studies at the college levelAbstract: The book, Introductory Statistics: Concepts, Models, and Applications, presented in the following pages represents...
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From the preface: "This is a book on linear algebra and matrix theory. While it is self-contained, it will work best for...
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From the preface: "This is a book on linear algebra and matrix theory. While it is self-contained, it will work best for those who have already had some exposure to linear algebra. It is also assumed that the reader has had calculus. Some optional topics require more analysis than this, however." A solutions manual to the exercises in the textbook is included.
According to The Orange Grove, "This book covers the following: Foundations of Trigonometry, Angles and their Measure,...
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According to The Orange Grove, "This book covers the following: Foundations of Trigonometry, Angles and their Measure, Applications of Radian Measure, Cosine and Sine, Beyond the Unit Circle, The Six Circular Functions and Fundamental Identities, Beyond the Unit Circle, Trigonometric Identities, Graphs of the Trigonometric Functions, Graphs of the Cosine and Sine Functions, Graphs of the Secant and Cosecant Functions, Graphs of the Tangent and Cotangent Functions, The Inverse Trigonometric Functions, Inverses of Secant and Cosecant: Trigonometry Friendly Approach, Inverses of Secant and Cosecant: Calculus Friendly Approach, Using a Calculator to Approximate Inverse Function Values, Solving Equations Using the Inverse Trigonometric Functions, Trigonometric Equations and Inequalities, Applications of Trigonometry, Applications of Sinusoids, Harmonic Motion, The Law of Sines, The Law of Cosines, Polar Coordinates, Graphs of Polar Equations, Hooked on Conics Again, Rotation of Axes, The Polar Form of Conics, Polar Form of Complex Numbers, Vectors, The Dot Product and Projection, and Parametric Equations. In addition, exercises and answers are provided for the reader.Most questions from this textbook are available in WebAssign. The online questions are identical to the textbook questions except for minor wording changes necessary for Web use. Whenever possible, variables, numbers, or words have been randomized so that each student receives a unique version of the question. This list is updated nightly.״
This is a free, open textbook that is part of the Connexions collection at Rice University. The book includes the following...
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This is a free, open textbook that is part of the Connexions collection at Rice University. The book includes the following topics: 1. Sampling and Data 2. Descriptive Statistics 3. The Normal Distribution 4. Confidence Interval 5. Hypothesis Testing 6. Linear Regression and Correlation. |
Avondale Estates AlgebraHere are topics for which you may want to hone your skills:
-Equations and inequalities
-Linear equations and functions
-System of linear equations and inequalities
-Matrices and Determinants
-Quadratic functions
-Polynomials and polynomial functions
-Powers roots and radicals
-Exponential and Lo... |
In MYP mathematics, the four main objectives support the IB learner profile, ... from their own and other subject groups teachers working in isolation multiple sources and resources for learning a textbook-driven curriculum students investigating, ...
The International Baccalaureate aims to develop inquiring, ... IBMaths helps the students to develop a variety of skills which include understanding, ... If a key point is not very clear then seek clarification from your notes or textbook.
Maths Assessment Guidelines. ... These can include questions from the textbook or homework book, test revision, ... In line with both the new International IGCSE Maths course and IBMaths courses, revision tests will consist of two main sections.
International Baccalaureate. Primary Years Programme. ... A choice of Chinese Maths or English Maths. ... PYP students access a wide range of learning resources and materials and do not follow one specific textbook. ( Collaborative learning.
In particular we are an International Baccalaureate world school. ... and the students start to use their GCSE textbook in Year 9. ... Mathematics is a very popular and successful subject in our Sixth Form with two thirds of our students taking A level Maths, Further Maths or IB options.
Students will follow advanced IB&M modules whose content extends beyond the basic textbook knowledge. The MSc in IB&M modules reflect state-of-the-art ... The MSc in International Business and Management, ... (IB/EB) that includes the subjects Maths Higher or Maths Standard/Methods and English, ...
[See also holdings for newer editions, if any]. Textbook recommended for a few of our courses/modules. May be out of print now. MMM114 Materials Science I. (a+, Oct) check on MMM Book List. MMM211 ... ENM101 Engineering MathsIB. (a, Oct) but check on ENM Book List. Books 035 MTL Record No ...
Mathematical Studies Standard Level for the IB Diploma. ... textbook exercise [ggb] GeoGebra activity [V] ... (This could also be used for a history of maths activity, as it contains a discussion of the historical development of 'zero'.) [www]
Mathematical Studies Standard Level for the IB Diploma. ... students should be encouraged to talk about these both in maths lessons and during specific Theory of Knowledge lessons. ... textbook exercise [ggb] GeoGebra activity [V] video link
... school, grades, maths. Easy! Choose the correct sources. Your essay must include a mixture of sources from: websites, books ... Keep in mind that the examiners are IB teachers from your chosen subject ... • Provided a foot note/ citation for any method found in a textbook or reference ...
... school, grades, maths. Easy! Choose the correct sources. Your essay must include a mixture of sources from: websites, books ... Keep in mind that the examiners are IB teachers from your chosen subject ... Provided a foot note/ citation for any method found in a textbook or reference ...
Should I Do More to Upgrade My Maths? What Study Skills Will I Need? ... e.g. IB) you will be taking ... For the revision of basic algebra, any GCSE (Higher Level) textbook will be useful, although Part 1 (chapters 1 to 5) of G. Renshaw. 2005. Maths for Economics, ...
in International Business endows its graduates with solid academic knowledge in Business Administration, ... Basic maths: introductory course. ... The students may choose among the following - There is no single adopted textbook.
Maths. Music. Physical Education. Science. ... A textbook will be used to resource the section on the French Revolution. Ancient Civilizations: ... An end of term assessment will follow the IB pattern.
There exist strong beliefs among them that if you do not follow the textbook, ... at the same time as some have great ambitions and want to enter the International Baccalaureate (IB) programme. ... Before, I had a hard time with maths, ...
Bored by Elizabeth I but could avoid questions on her in the exam. European History – big syllabus (as for IB today but not modern A levels) - included ... Which would be in a modern textbook. ... So I never got on with maths because I used to say, 'Hang on a minute, I need to know ...
The existing curriculum has yet to be fully integrated into textbook content, teacher training, and student assessment or to be developed in detail in the teachers' guides to textbooks. In addition, ...
Elements was widely considered the most successful and influential Mathematics textbook of all time, ... A complex number a + ib represents a point in a plane. ... Russian maths genius Perelman urged to take $1m prize bbc.co.uk, Wednesday, 24 March 20102.
The information contained in these notes is correct to my knowledge and by no means copied from any textbook, therefore, ... The (.ib model for the voltage controlled current source comes in useful for the 'common ... This time I'll go through the maths a lot faster, and let you work it out ...
... e.g. a textbook, ... Maths and Science, Modern Foreign Languages, Humanities, Art, ... In addition, we are the only college in St Helens offering the highly regarded International Baccalaureate (IB) Diploma. At Cowley we strive to create a partnership between students, parents, ...
... asked the class which of us enjoyed Maths and preferred translating from English to Latin rather than from Latin ... I had been amazed by a sentence in my history textbook mentioning Julius Caesar's De Bello Gallico as something `you ... But this is an insidious and damaging course.(ib.).
Textbook in preparation where nuclear physics and particle physics in integrated. ... THE PROFILE OF OUR SCHOOL IS MATHS-SCIENCES OVER ALL THE FOURTH GRADES. Several times, with several persons ... More of Particle Physics I teach within IB Diploma Programme.
1 Carnegie unit in an IB course with a score of 4 or higher on the exam OR. ... (English I, English II, 2 required maths, 2 required sciences, 2 required social studies ... and current events related to astronomy. The course work will include textbook assignments, Internet activities, online ...
... the IB Middle Years and Diploma programmes and the Pre-U examination. ... other than in the core subjects of maths, English and science, ... or superior to the exemplar material in the course textbook.
3.21 The International Baccalaureate includes an extended research essay as a final year capstone to the ... Communicating Maths is an optional module for third and fourth year students in the ... It builds on the more textbook-orientated knowledge and limited controlled laboratory ...
The International Baccalaureate Program is a rigorous academic program for students in their final two years of high school. ... Biology HL Group 5 Maths and computer science. Math Studies SL. ... Beside the textbook, students will use resources such as magazine articles, songs, games, ...
In this case the text will be heavily referred to but will not be the only textbook formally used by the lecturer in the delivery of the course. ... for example lecture material (maths and physics), ... Mathematics IB. Laboratory I Electricity. Thermodynamics. Mathematics II. Laboratory II ...
Research project on human resource management at Shanghai University of International Business and Economics (SUIBE), China. ... It builds on the more textbook orientated knowledge and limited controlled laboratory experiences in years one and two. |
Algebra and Trigonometry - 2nd edition
ISBN13:978-0201525106 ISBN10: 0201525100 This edition has also been released as: ISBN13: 978-0201640717 ISBN10: 0201640716
Summary: Interested in promoting interaction in the classroom? Then take a look at the Bittinger/ Beecher, Algebra and Trigonometry, Second Edition. This paperback text is designed specifically to motivate your students to participate actively and immedately. This paperback text is crafted to meet the varied skill level of your students-giving them the solid content coverage they need in a supportive format. Plus, these texts foster conceptual thinking with thinking and writi...show moreng exercises, computer/graphing calculator exercises, and a thoroughly integrated five-step problem solving approach. This worktext features a right triangle introduction to trigonometry. ...show less26 |
Young scholars investigate geometric series. In this Algebra II/Pre-Calculus lesson, students use a spreadsheet application to create partial sums of geometric series and observe convergence and divergence. Young scholars investigate the relationship between the behavior of the graph and the common ratio of the series. Students examine the formula for the sum of a divergent infinite geometric series.
Students find patterns in a sequence. In this sequences and series lesson, students use their calculator to find the sequence of partial sums. They graph functions and explore convergent series. Students approximate alternating series.
In this infinite geometric series worksheet, 11th graders solve and complete 33 various types of problems. First, they find the sum of the infinite geometric series for each of the given. Then, students find the range of values for each of the following infinite geometric series converge to. They also evaluate each of the sums in sigma notation.
In this Algebra II instructional activity, 11th graders use their knowledge of infinite geometric series to solve problems such as determine in the sum of a series, solving word problems, and converting repeating decimals to fractions. The one page instructional activity contains sixteen problems. Answers are not included.
Learners analyze geometric series in detail. They determine convergence and sum of geometric series, identify a series that satisfies the alternating series test and utilize a graphing handheld to approximate the sum of a series.
Looking for nice real-life example of a finite geometric series? In this video, a geometric series is used to calculate the monthly payment amount on the fixed rate loan. Starting with a general equation of monthly payments, Sal shows how the calculation of payments is actually a geometric series. He then uses this series to calculate an example payment for a loan.
In this sequences and series activity, 10th graders solve and complete 13 different problems that include infinite geometric series. First, they find the sum of each infinite geometric series. Then, students find the first three terms of each geometric series. They also express each decimal as a rational number in a given form.
Your mathematicians solve problems using geometric series in this pre-calculus lesson. They divide a unit square into very small pieces and identify the reasoning behind the sum of a series being less than one.
Learners will determine if a series of numbers are geometric, arithmetic, or neither and determine the common ratio if appropriate. In this geometric units lesson, students given a geometric series, determine the general term, and the general sum.
In this Algebra II/Pre-calculus worksheet, students solve problems that involve arithmetic series and infinite geometric. Students determine the partial sums of a geometric series. The three page worksheet contains forty-six problems. Answers are not provided.
Students determine the sum of an infinite geometric series. In this determining the sum of an infinite geometric series activity, students discuss how the sum of an infinite geometric series is a paradox. Students use the sum of an infinite geometric series formula to show that repeating decimals have a fractional representation. Students solve paradox problems using the sum of infinite geometric series formula.
Why should one take the whole regimen of prescribed antibiotics? Use forms of a finite geometric series to find out. Change from expanded form to closed form in the geometric series and make calculations of the amount of antibiotics in the body at a given time. Graph the calculations and analyze the data to draw conclusions.
Real-world application of the finite geometric series. Sick? Need to take Ampicillin? How much is in the body at any given point in time as this medication is taken over the course of time. Use the formulas, make calculation, and graph the findings to draw conclusions.
In this interval worksheet, students write a geometric series for the derivatives of a function. Students determine the values of x and where it may converge. This two-page worksheet contains five multi-step problems. |
"In statistics, there are many discrete probability distributions. The binomial distribution is a discrete probability
distribution that describes the probability of success in a binomial experiment. Learn more with CK-12 Flexbooks"
"CK-12 Flexbooks on Parallelograms explains there are two main types of quadrilaterals: parallelograms and non-parallelograms. Parallelograms can also be divided
into many sub-groups, including rectangles, trapezoids, and squares. Each type of parallelogram has its own unique
set of properties."
"CK-12 Flexbooks on Polygon explains Polygons consist of line segments connected only at their endpoints. Many types of polygons exist, with varying side
and angle measurements. Using formulas, we can find the sum of the interior angles of a polygon."
"CK-12 Flexbooks on Ratios and Proportions explain The use of ratios and proportions comes in handy when finding the relationship between two geometric shapes. Ratios
and proportions are often an important part of certain properties between two shapes, such as two similar polygons."
"CK12 Flexbooks on Properties of exponent explains
When an exponent is a positive whole number, an exponent is a shorthand way to represent large quantities of
multiplication. Exponents can represent any value that is multiplied by itself repeatedly (for example, an expression
such as 5·5·5·5). Variables can have exponents just like real numbers"
"CK12 Flexbooks on Operations with polynomial explains Monomials and polynomials can contain numbers, variables, and exponents. They can be added, subtracted, multiplied,
divided, and factored, just like real numbers. There are a few special products of polynomials that are important to
know, such as the product of two binomials." |
Mathematics
About courses in Mathematics.
Courses and Descriptions
MA-01 Getting Ready for Algebra by Using Virtual Manipulatives
This course will prepare teachers to use virtual manipulatives to help their students get ready for algebra. Participants will explore the 17 Algebra Readiness indicators developed by the SREB (Southern Region Education Board) and ETS (the Educational Testing Service), including the five "process" indicators and the twelve "content and skills" indicators. Participants will learn how they can use virtual manipulatives and other technologies to help their students gain proficiency in order to be successful algebra students. Participants will complete a technology enhanced classroom project for their students that is aligned to NCTM and state standards.
Syllabus | Rubric
Addressing the individual learning styles of students can be a challenge for teachers. The World Wide Web contains a vast number of resources to assist teachers in understanding and planning for the different avenues through which students learn best. Suitable for participants of all grade levels and subject areas, this course will review a range of web sites providing information about learning theory related to learning styles and multiple intelligences, as well as resources to assist teachers in both identifying students' learning styles and intelligences and engaging students in activities which best suit those styles and intelligences. Participants will become familiar with teaching strategies and tools targeted for each learning style and intelligence and develop a preliminary lesson plan using those strategies and tools.
Syllabus | Rubric
MA-03 Using Patterns to Develop Algebraic Thinking
In this course for middle school teachers of mathematics, participants will explore the nature of algebraic thinking and develop strategies for encouraging the development of algebraic thinking in their students. Through readings, video clips, and examination of student work, participants will learn to: 1) recognize and build on opportunities for algebraic thinking in a variety of mathematics contexts; 2) analyze their students' algebraic thinking; and 3) pose questions that encourage the development of algebraic thinking. Participants will also produce a lesson plan that incorporates the key concepts of the course.
Syllabus | Rubric
MA-04 Using Real Data in Math Classrooms
Technology tools and web-based materials provide important ways for math educators to meet local and national standards that emphasize problem solving and making connections between mathematics, other disciplines and the real world. This course will enable middle and high school math teachers the opportunity to explore a range of web-based resources and exemplary projects which utilize technology to support these goals. Participants will learn how to find sources of real data on the web and explore technology tools that help students model, analyze, visualize and make sense of these data. Participants will complete the course with a collection of resources and beginning project ideas that serve their curricular goals.
Syllabus | Rubric
MA-05 Using Technology in the Elementary Math Classroom
In this course, participants will explore technologies that can be used in elementary math instruction in kindergarten through sixth grade. Participants will review NCTM and state standards and examine the ways in which tools like virtual manipulatives, calculators, spreadsheet programs, online data sources, and applets can support these goals. In particular, the course will address ways in which technology can support elementary algebra, geometry, and data analysis standards. Participants will leave the course with complete lesson plans for integrating technology into instruction in their own classrooms.
Syllabus | Rubric
MA-06 Using Models to Understand Fractions
This course provides an opportunity for teachers of grades 3 through 5 to explore how mathematical models can be used to promote a deep understanding of fractions and fractional relationships. All too often, students are taught mechanical ways to make sense out of fractions, whether in the form of an addition algorithm or a procedure that produces equivalent fractions. This course offers alternative ideas for fraction instruction and is premised on the belief that students can come to understand fractions in a number of different ways. Completion of a student interview and a final project are integral parts of this course. A variety of readings, applets, and videos form the content of this course, and participants are expected to share ideas with their online colleagues in the discussion forums.
Syllabus | Rubric
MA-07 Algebraic Thinking in the Elementary School
This course will enable you to learn how to include opportunities in the upper elementary curriculum for children to think about basic algebraic concepts, and to appreciate the importance of algebraic thinking. You will read and discuss relevant research on the importance of algebraic thinking in elementary school instruction and explore a variety of problems that can be used with your students to develop their algebraic thinking. You will investigate student misconceptions about the sign "=" and why this is such a pivotal concept in elementary mathematics. For the final project, you will learn how to design and "algebrafy" activities which encourage algebraic thinking that can be immediately integrated into your classroom practice.
Syllabus | Rubric
MA-08 Proportional Reasoning
In this course, you will learn about different components and levels of proportional reasoning. You will complete classroom tasks that illustrate ways to help your students develop stronger conceptual understanding about the components of proportional reasoning. You will explore both video and written examples of student work and will conduct a student interview in order to reflect on how students approach proportional reasoning and how to identify evidence of proportional reasoning among students. For the final project, you will create a lesson that targets a component of proportional reasoning and incorporates questioning techniques learned in the course. You will also provide a rationale for the lesson that relates to the material learned in the course and reflect on your own learning.
Syllabus | Rubric
MA-09 A Conceptual Introduction to Functions: Using Visual Models
In this course, you will learn strategies to help students in grades 6-8 develop a strong conceptual understanding of function through the use of several different visual models. You will become familiar with common student misconceptions related to functions, as well as classroom tasks that address these misconceptions. You will complete and create classroom tasks that will allow your students to make connections between multiple representations of functions and to explore both linear and non-linear functions. In addition, you will explore the use of technology in supporting student learning about functions and will learn to use an analysis tool to evaluate various online applets. Your final project will be to develop a classroom activity that incorporates the use of technology focused on students' conceptual understanding of function, and you will reflect on your learning in the course.
Syllabus | Rubric
MA-10 Promoting Mathematical Discourse in Grades K-4
Ask a group of elementary school students to name a language, and chances are they will not reply "mathematics." Yet from the time young children begin going to school, they are expected to discuss their mathematical thinking using the vocabulary, structures, and ideas of mathematics. In this module, participants will learn why communicating about mathematics is important for young students, as well as why it can be so difficult for them. They will consider how math and literacy instruction intersect in the elementary classroom, and explore ways of pushing students to explain their thinking beyond "I just know the answer." Participants will learn questioning strategies that can help students communicate, justify, and defend their mathematical ideas, and watch videos of real classroom teachers using these strategies. By the end of this module, participants will have created a lesson plan that integrates communication-based activities into their elementary math instruction.
Syllabus | Rubric
online professional development to meet your needs
Online Course Model
Online Course Orientation All OPEN NH courses begin with an optional face-to-face orientation, allowing participants to become familiar with course login procedures and overall course requirements. Orientation to the course continues online during the first week, so that participants become familiar with course expectations and get acquainted with their instructor and fellow classmates.
Online Content and Practice Following the orientation, courses continue for six weeks, emphasizing theory-to-practice applications. Course work includes online discussions, reflections, readings, and projects. Course participants receive a certificate of participation that may be applied to their individual professional development plans.
Registration Register for courses by visiting the Registration link in the Navigation bar.
Early Bird registration special apply up to two weeks before course session begins.
If you plan to register in the last week before courses start, contact the e-Learning for Educators Project
, or visit the e-Learning Blog, to find out the course status. Not all courses may run.
On Your Own e-Learning While our regular course registration fee is $130, we also offer self-paced tutorials at even lower cost. These tutorials have an assessment component and will earn you professional development CEUs after successful completion. District cohort deals are also available. Details on the Self Paced Tutorials page.
NH Department of Education Tutorials The New Hampshire Department of Education is committed to providing guidance and service to schools and districts. OPEN NH offers free online tutorials that support NHDOE initiatives and programs.
Institutes in a Box As part of the NH Digital Resources Consortium, OPEN NH is hosting the online workshops and Institutes.
Coming Soon.
Conservation Education As part of the NH Environmental Literacy Plan implementation, OPEN NH is hosting freely accessible tutorials on Conservation Education.
Free Tutorials There are also free tutorials available, which do not earn CEUs. These will help introduce you to the online learning environment and introduce you to our course model and content. Click on the Moodle Course Login link in the Upper Right hand corner and select "Self Paced Tutorials". |
Get the confidence and math skills you need to get started with calculus Are you preparing for calculus? This hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in the course. You'll get hundreds of valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step |
If you or someone you know needs help, let me know!Algebra 1 topics include: solving and graphing linear equations, basic operations with algebraic expressions, polynomials, exponents, and applications of all of the above. Algebra 2 covers all the material from Algebra 1 but more in depth, along... |
What KSAs (knowledge/skills/abilities) are tested (e.g., is cultural diversity included)? The test is intended to focus on problem solving, communication, reasoning, and mathematical connections. "This basic mathematics test requires the examinee to demonstrate an understanding of basic concepts and their applications in constructing a model, writing a proof, and solving two problems" (Tests at a Glance: High School Mathematics, p. 36). The basic mathematics content in this test covers arithmetic and basic algebra, geometry, analytical geometry, functions and their graphs, probability and statistics (without calculus), and discrete mathematics. The test assesses knowledge in at least five of these six areas. Each of these six areas is described in some detail below.
To solve the four problems, examinees must be able to understand and work with mathematical concepts, reason mathematically, integrate knowledge of different areas of mathematics, and develop mathematical models of real-life situations (ETS, June 2000, Test Analysis Subject Assessments Mathematics: Proofs, Models, and Problems, Part 1).
Comment: The topical coverage of the test seems reasonable when considering that it is only one of several tests in a series of mathematics tests (but a content specialist could judge more appropriately the quality and completeness of the content coverage). There is no indication of what percentage of the test is related to each content area or to the broader skills (e.g., work with mathematical concepts).
There is also a caveat in the Tests at a Glance: High School Mathematics that examinees may have to draw on competencies from other content areas to solve problems. It is not clear if this means other areas of mathematics (e.g., calculus) or other content areas such as history.
The absence of a more specific distribution of test content and the potential for the examinees to need skills other than those specified in the preparation materials are problematic. The first problem can lead to validity concerns, especially considering the comparability of scores across forms. The second problem suggests the potential for validity problems in terms of potential contamination of scores due to the need for skills other than those for which inferences from the scores may be desired.
How were the KSAs derived and by whom? The content domain1 was determined by using a job analysis procedure that began in 1989.
1
This domain was for all tests that might be used to license teachers of mathematics; thus, it is broader than the domain for this test, which is only one test out of six that relate to high school mathematics. The total domain contains 12 major knowledge categories: Basic Mathematics, Geometry, Trigonometry, Functions and Their Graphs, Probability and Statistics, Calculus, Analytical Geometry, Discrete Mathematics, Abstract Algebra, Linear Algebra, Computer Science, and Pedagogy Specific to Mathematics. |
If you think algebra has to be boring, confusing and unrelated to anything in the real world, think again! Written in a humorous, conversational style, this book gently nudges students toward success in pre-algebra and Algebra I. With its engaging question/answer format, step-by-step explanations, lively cartoons, and practice problems for each new concept, the Guide helps struggling students have fun learning algebra. The Algebra Survival Guide's 12 content chapters tackle all the trickiest topics: Properties, Sets of Numbers, Order of Operations, Absolute Value, Exponents, Radicals, Factoring, Cancelling, Solving Equations, the Coordinate Plane and yes even those dreaded word problems. The Guide is loaded with practice problems and answers, and its 288 pages give students the boost they need to master the skills of algebra.
Illustrations by award-winning Santa Fe artist Sally Blakemore help deepen understanding for visual learners and make the learning process entertaining for everyone!
Additional Information
Weight
1.95 lbs
Dimensions
11 x 8.5 in
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The Matrix Algebra Tutor: Learning by Example DVD Series teaches students about matrices and explains why they're useful in mathematics. This episode teaches students about the inverse of a matrix, including when its used to solve systems of equations, and how to compute the inverse of a matrix by hand. Grades 9-College. 51 minutes on DVD. |
Take it step-by-step for math success!
The quickest route to learning a subject is through a solid grounding in the basics. So what you won?t find in Easy Mathematics Step-by-Step is a lot of endless drills. Instead, you get a clear explanation that breaks down complex concepts into easy-to-understand steps, followed by highly focused exercises... more...
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This volume supports the belief that a revised and advanced science education can emerge from the convergence and synthesis of several current scientific and technological activities including examples of research from cognitive science, social science, and other discipline-based educational studies. The anticipated result: the formation of science... more...
CliffsQuickReview course guides cover the essentials of your toughest subjects. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. Whether you're new to limits, derivatives, and integrals or just brushing up on your knowledge of the subject, CliffsQuickReview Calculus can help. This guide covers... more...
This unique resource provides 190 high-interest, ready-to-use activities to help students master basic math skills— including whole numbers, decimals, fractions, percentages, money concepts, geometry and measurement, charts and graphs, and pre-algebra— for use with students of varying ability levels. All activities are classroom-tested... more...
Blackline master math activity book jam packed with adventures
Blackline master math activity book jam packed with
adventures |
look forward to hearing from you.Algebra 1 is the study of elementary algebra. It is the next step, after pre-algebra, in the development of higher level mathematics skills. Algebra 1 emphasizes equation problem solving and graphing. |
Book summary
This text presents the principal ideas of classical number theory emphasizing the historical development of these results and the important figures who worked on them. It is intended to introduce third or fourth-year undergraduates to mathematical proofs by presenting them in a clear and simple way and by providing complete, step-by-step solutions to the problems with as much detail as students would be expected to provide themselves. This is the only book in number theory that provides detailed solutions to 800 problems, with complete references to the results used so that the student can follow each step of the argument. [via] |
Mathematical Ideas - 11th edition
Summary: One of the biggest issues college math instructors face is capturing and keeping student interest. Over the years, John Hornsby has refined a creative solution--bringing the best of Hollywood into his mathematics classroom.Mathematical Ideas applies this same strategy of engaging students through video clips from popular cinema and television to the textbook. Alongside fresh data and tools, this Edition uses up-to-the-minute images as well as old favorites of math be...show moreing done in Hollywood. In addition, examples are clarified with additional annotations, chapter summaries are made more intuitive to aid review, and chapter tests now include specific section references, making it easier for students to refer back to topics that need more attention. With great care and effort, the authors have crafted this new edition to serve the needs of today's students and instructorsAcceptable
Borgasorus Books, Inc. MO Wentzville, MO
Hardcover Fair 0321361482 Student Edition. Missing many pages. Heavy wear, wrinkling, creasing, Curling or tears on the cover and spine May be missing front or back cover. May have used stickers or...show more residue. Fair binding may have a few loose or torn pages. Heavy writing, highlighting and marker. ...show less
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Goodwillnyonline Astoria, NY
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Follett School Solutions, Inc. Woodridge, IL
0321361482 |
Description The first in a series of two courses designed to provide an in-depth study of algebra to prepare students for advanced mathematics courses. Graphical analysis of concepts is emphasized through the use of technology. Topics include working with algebraic expressions, solving equations algebraically and graphically, and a detailed analysis of the algebraic and graphical properties of various functions. Prerequisite: Grade of C or higher in MATH 080F, appropriate score on placement test or permission of the Mathematics Department. Formerly MATH 109, Precalculus I. |
Description
Taking a starting point below that of GCSE level, by assuming no prior mathematical knowledge, Surinder Virdi and Roy Baker take the reader step by step through the mathematical requirements for Level 2 and 3 Building and Construction coursesConstruction Mathematics |
Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology... more...
The fundamental aim of this book is to communicate the knowledge necessary for a competent mathematical assessment of modern developments in the age of the Internet. Most crucially, this includes an understanding of very large graphs, calculating with large numbers, and calculating using prime number bases. more... |
A series of online tutorial units with quizzes designed for students in the introductory college or university physics course. The page includes topics from math, physics, and biophysics. Remedial material includes...
Graph is "an open source application used to draw mathematical graphs in a coordinate system." Anyone who wants to draw graphs of functions will find this program useful. The program makes it very easy to visualize a...
This unit from Illuminations focuses on collecting data and using technology to find functions to describe the data collected. Students will learn to use a calculator to find the curve of best fit for a set of data and...
This lesson from Illuminations looks at inverse variation. Students are asked to use estimates to measure the height of an object, visualize two-dimensional objects and calculate their area, and sketch graphs to... |
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Available Resources
Videos: Explanations of some of the most commonly used features and functionality.
Tip Sheets: Detailed instructions to help you get the most out of MathXL for School.
Help System: Once you have registered and signed in to MathXL for School, be sure to look for the help resources available within the product:
Teachers & Students: Look for the ? symbol in the upper right corner of nearly every page for detailed information about the page you are viewing.
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Self-Study and Training Certification: Designed to teach you how to successfully use MathXL for School to its maximum effectiveness, these courses allow you to work through online courses to better understand the program.
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MTTC 101 MathXL – Beginning This course is designed to give you an overview of MathXL for School from a student's perspective, as well as instruction in setting up your first MathXL for School course.
MTTC 201 MathXL – Intermediate This course will focus on teaching strategies using MathXL for School. In particular, you will learn how to use and customize homework assignments, tests, and the gradebook, and manage multiple sections of a course. PREREQUISITE: MTTC 101.
MTTC 204 Custom Question Builder – Introduction This course will introduce participants to the Custom Question Builder (CQB). This innovative tool is available in MathXL for School, and it allows you to create your own custom exercises for use in homework assignments, quizzes, or tests. PREREQUISITE: MTTC 201 or equivalent. One or more terms of experience teaching with MathXL for School is preferred.
MTTC 304 Custom Question Builder – Advanced This course will show you more tools to use to create custom exercises in the Custom Question Builder (CQB), such as adding number lines, graphs, pie charts, bar charts, and graphics. PREREQUISITE: MTTC 204 or equivalent. |
Fundamental Theorem of Algebra
Description: The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. TheMore...
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The fundamental theorem of algebra states that any complex polynomial must have a complex root. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs leads to more general results from which the fundamental theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair. To arrive at each of the proofs, enough of the general theory of each relevant area is developed to understand the proof. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the transcendence of e and pi are presented. Finally, a series of appendices give six additional proofs including a version of Gauss'original first proof. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students, and would make an ideal "capstone" course in mathematics |
This book contains short lessons on topics in math and physics, written in a style that is jargon-free and to the point. Each lesson covers one concept at the depth required for a first-year... More > university-level course. The main focus of the book is to show the intricate connections between the concepts of mechanics and calculus. Often calculus and mechanics are taught as separate subjects. It shouldn't be like that. If you learn calculus without mechanics, it will be boring. If you learn mechanics without calculus, you won't truly understand. This textbook covers both subjects in an integrated manner and aims to highlight the connections between them.
In order to make learning calculus and mechanics more accessible, the book begins with a review chapter on numbers, algebra, equations, functions, and other prerequisite concepts from high school math. Anyone should be able to pick up this book and become proficient in calculus and mechanics, regardless of their mathematical background.< Less |
Math Triumphsis an intensive intervention resource for students who are two or more years below grade level. The series accompanies GlencoeAlgebra 1,Geometry, andAlgebra 2and provides step-by-step intervention, vocabulary support, and data-driven decision making to help students succeed in high school mathematics.. |
Math Cove projects include: Java applets for exploring transformations (translations, reflections, and rotations) and learning about acute and obtuse angles and other concepts in triangle geometry; more than 20 lessons for studying graph theory (using the software package Petersen); and an applet for studying tiling with right trominoes. With Instructors' Notes that discuss the uses of Java applets in the math classroom. Java source files and comments are supplied, along with an online paper, "Uses of Java Applets in Education." |
MATH121
Every aspect of Math 121 is highly interactive: Students spend most of classtime working in groups on problems and they then present their work and discuss as a class. Each student is responsible for some part of the in-class problems. Read more about Interactive Group Math |
·MATH 648K (Fall
2011)
·The principle
of minimal action in geometry and dynamics
·Instructor: VadimKaloshin
MATH 3111,
vadim.kaloshin@gmail.com, (301) 405-5132.
Course
Information
The course will be devoted to analysis
of various minimization procedures in geometry and Hamiltonian dynamical systems
and their deep relation. For example, dynamics of planar billiards in convex
domains is closely related to the famous Kac's
question: Can you hear the shape of a drum? |
SpeQ Mathematics 3.3 description
SpeQ is a small, extensive mathematics program with a simple, intuitive interface. All calculations are entered in a sheet. In there you can freely add, edit and execute all calculations. You can define variables and functions, and plot graphs of your functions. You can save your calculations for later re-use.matlab 6.1
Features:
SpeQ has the following features.
Expression calculations with arithmetic, conditional, logical and bitwise operators
Around 100 built-in mathematical, physical and conditional constants
Around 60 built-in functions for Analysis, Arithmetic, Complex numbers, Hyperbolic, Integers, Logical, Probability, Numeral systems, Statistics, Trigonometry
Define your own variables and functions for use in calculations
Plot fully adaptable graphs, trace the plotted functions, save your graphs
Extensive support for units
Decimal, binary, hexadecimal and octal numeral systems
Calculations with lists
Complex number calculations
Colored workarea
Save and load sheets
Orderly overview of all available functions and constants in the Functionstree
A list with all defined variables and functions in the Memorylist
Context-sensitive help and a detailed users guide
SpeQ Mathematics 3.3
Advanced yet easy-to-use math calculator that immediately and precisely computes the result as you type a math expression. It allows multiples math expressions at same time. It also allows fractions a Free Download |
Algebra : Form and Function (Preliminary Edition) - 08 edition
Summary: This text covers all of the standard topics for college algebra. The first four chapters give an introduction to algebra for those students who need it. There is also a cumulative review exercise at the end of Chapter 4.The exercises are a normal lesson apart, and the problems in each exercise are in groups of four similar ones. This makes it a simple matter for even the inexperienced instructor to make a good assignment regularly. Most classes only need to be assign...show moreed every fourth problem, but other problems are available for practice. There are about 5000 problems in some 75 regular and 12 review exercises. About half of the problems are new, and there are many drill problems which are closely keyed to the examples.Answers are given in the text for three-fourths of the regular and all of the review problems04 |
* This product is not available for shipment to certain countries. We apologize for any inconvenience.
Product Description
Students will learn the basics of higher-level math concepts, focusing on sets & logic, with School of Tomorrow's Basic Mathematics I curriculum. PACE Self-Pac 2 covers elements of a power set, intersection and union, defining and identifying disjoint sets, defining the complement of a set, defining the relative complement of a set with respect to another set, to define unary and binary operations, and to perform operations in the right order when an expression is punctuated with brackets.
Fill-in-the-blank, multiple-choice, and write-in comprehension exercises help with review and retention. Paperback booklet. This resource is listed as "college" or post-secondary curriculum; parents may wish to consider for grades 8 & up.
Please note that the not-included book "Man's Mathematical Models" by Bill Williams and Gwen Crotts is a required resource for this course |
The purpose of the book is to give the reader a feeling for the beauty and the surprises of mathematical research by building up step by step a theory of cycle matrices. The notions of matrix and cycle are presented for those readers who do not yet know the concepts. Then, starting and experimenting with cycle matrices of low dimensions and engaging the readers in various activities, beautiful results are obtained with attractive geometrical interpretation. Gradually the results are extended to cycle matrices of higher dimensions and then generalised. A note is included for those readers who would like to use a computer in exploring cycle matrices. Two appendices give the possibility to further use the computer by exploring inverses matrices and determinants.
I highly recommend this Dover text as either a self-tutorial or as a quick review of matrices. I initially thought that a text for the "non-mathematics science student" might prove too elementary to be useful. It actually provides a thorough, concise (but not terse) introduction to matrices that is quite appropriate for undergraduate students in physics or chemistry. The sections on eigenvalues and eigenvectors, on diagonalization of matrices, and on functions of matrices were particularly good.
A matrix is an ordered set of numbers listed in rectangular form having certain rows and columns. The knowledge of matrices is considered necessary in various areas of Mathematics and a wide variety of other subjects, like Chemistry, Sociology, Genetics and Electrical Engineering. The importance of Matrix Theory has greatly increased with the invention of Electronic Brain—Computer. It has become a powerful tool of modern Mathematics and is now extensively used in the allied fields of Mathematics including Probability, Statistics, Quantum Mechanics, Mechanics, etc. A Textbook of Matrices covers all the important topics on matrices.
This definitive presentation of matrix theory is the work of one of America's most significant contributors to this area. The first three chapters take matrix theory up to elementary divisors, proceeding to examinations of vector polynomials and singular matric polynomials. A discussion of compound matrices follows, succeeded by considerations of Hermitian, symmetric, and skew matrices, with a number of interesting proofs of properties. Subsequent chapters are devoted to commutative matrices, matrix functions, and the automorphic transformation of a bilinear form. The final chapter offers a masterly handling of linear associative algebras. ...
This text features material of interest to applied mathematicians as well as to control engineers studying stability of a servo-mechanism and numerical analysts evaluating the roots of a polynomial. Includes complex symmetric, antisymmetric, and orthogonal matrices; singular bundles of matrices and matrices with nonnegative elements. Also features linear differential equations and the Routh-Hurwitz problem. 1959 edition. ...
This book develops the subject of matrices with special reference to differential equations and classical mechanics. It is intended to bring to the student of applied mathematics, with no previous knowledge of matrices, an appreciation of their conciseness, power and convenience in computation. Worked numerical examples, many of which are taken from aerodynamics, are included. ...
A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the last decade. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems.
Random matrices are widely and successfully used in physics for almost 60-70 years, beginning with the works of Dyson and Wigner. Although it is an old subject, it is constantly developing into new areas of physics and mathematics. It constitutes now a part of the general culture of a theoretical physicist. Mathematical methods inspired by random matrix theory become more powerful, sophisticated and enjoy rapidly growing applications in physics. Recent examples include the calculation of universal correlations in the mesoscopic system, new applications in disordered and quantum chaotic systems, in combinatorial and growth models, as well as the recent breakthrough, due to the matrix models, in two dimensional gravity and string theory and the non-abelian gauge theories. The book consists of the lectures of the leading specialists and covers rather systematically many of these topics. It can be useful to the specialists in various subjects using random matrices, from PhD students to confirmed scientists.
Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book's primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted.
This text stresses the use of matrices in study of transformations of the plane. Familiarizes reader with role of matrices in abstract algebraic systems and illustrates its effective use as mathematical tool in geometry. Includes proofs of most theorems. Answers to odd-numbered exercises. |
...
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This Book
so that they will better understand and apply the skills outlined in the Standards.
This important resource also gives teachers a wealth of tools and activities that can encourage students to think critically, use mathematical reasoning, and employ various problem-solving strategies.
Filled with activities that will help students gain an understanding of math concepts and skills correlated to the Common Core State Math Standards
Offers guidance for helping students apply their understanding of math concepts and skills, develop proficiency in calculations, and learn to think abstractly
Describes ways to get students to collaborate with other students, utilize technology, communicate ideas about math both orally and in writing, and gain an appreciation of the significance of mathematics to real life
This practical and easy-to-use resource will help teachers give students the foundation they need for success in higher mathematics.
Related Subjects
Meet the Author
Judith A. Muschla has taught mathematics in South River, New Jersey, for over twenty-five years. She has been a member of the state Standards Review Panel for the Mathematics Core Curriculum Content Standards in New Jersey.
Gary Robert Muschla taught at Appleby School in Spotswood, New Jersey, for more than twenty-five years. Judith and Gary have coauthored several books, including The Algebra Teacher's Activities Kit, Hands-On Math Projects with Real Life Applications, The Math Teacher's Book of Lists, and Math Starters.
Erin Muschla teaches 8th grade math and algebra at Monroe Township Middle School in Monroe, New Jersey, and has collaborated with Judith and Gary on three previous math |
The Zero the Math Hero - Geometry Tutor DVD Series teaches students through step-by-step example problems. This DVD teaches students four important theorems and describes the Pythagorean Theorem and its relevance in the real world. Problems using algebraic skills are linked with geometric concepts. Students will learn the Pythagorean Theorem, the converse of the Pythagorean Theorem, the Isosceles Triangle Theorem, and the converse of the Isosceles Theorem. Grades 8-12. 22 minutes on DVD. |
An instructor at Boise State University offers his book on elementary algebra for middle school, high school, and college students. The book spans nine chapters and covers both pre-algebra and algebra topics. Clear excellent metasite for prime numbers is mind-boggling, not only in terms of the quality of information provided, but also in the breadth and variety of the hyperlinks, from discourse on the Riemann hypothesis toDoes knowing about Descartes help students understand the Cartesian coordinate system? Will teaching students the origin of the term parabola help them understand the mathematical importance of parabolas and other... |
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MATH - Mathematics
This course includes topics such as polynomials, radicals, exponents, coordinate geometry, rational expressions and equations, and solutions to linear and quadratic equations. Students are introduced to the concept of functions and their graphs. Additional topics may include conic sections, matrices, variation, and nonlinear inequalities. Emphasis will be placed on problem solving. A graphing calculator is required. The course is designed to give students additional time above that allotted in MATH 1033 working on mastery of concepts and skills in the student learning outcomes.
This is the beginning study of the solution of differential equations with emphasis on both analytic and numerical solutions. Topics include first and second order differential equations and their solutions, series solutions, Laplace transforms, linear equations of higher order, numerical solutions or ordinary differential equations using Euler and Runge-Kutta methods, and the use of Eigenvalue methods to solve linear systems. In addition, this course emphasizes the development of differential equations as mathematical models for a variety of practical applications.
This course is designed as a continuation of MATH 2094. Topics will include: parametric equations, polar, cylindrical and spherical coordinate systems, vectors and vector valued functions, functions of several variables, partial derivatives and applications, multiple integrals, and vector analysis, including Green's theorem, Stokes' theorem, and Gauss' theorem. The course will include several major projects outside of class.
This course will introduce the students to the following topics: order of operations, operations on real numbers, simplifying algebraic expressions, integer exponents, solving linear equations in one variable, graphing linear equations in two variables, and applications such as geometry and modeling. Emphasis is placed on reviewing basic arithmetic skills and elementary algebra topics. Development of arithmetic skills throughout the semester is essential, therefore students will not be allowed to use calculators.
This course is intended for students who need more preparation to be successful in College Algebra or other courses of that level. Topics covered include: review of first degree equations, systems of equations and inequalities, graphing, polynomials, factoring, radicals and rational exponents, quadratic equations, rational expressions, relations and functions and an introduction to triangle trigonometry.
This calculus-based course offers the theoretical basis for probability and statistics related to engineering applications. Topics include data analysis techniques, random variables, expectation, important probability distributions and densities, inferences concerning one or more means and standard deviations. Reliability, correlation and regression, curve fitting, and quality control charts are introduced. Graphing calculators are required. Computer applications may be included.
This course is designed for the engineering technology student. It covers techniques for comparing alternative projects based on economic considerations; time value of money; present worth; equivalent uniform annual cost; rate of return on investment; minimum cost life; expected value; decisions under risk; effects of income tax and inflation.
A student may contract from one to four credit hours of independent study in mathematics through an arrangement with an instructor of mathematics. The student and instructor will develop a course of study which must be approved by the department chairperson and the school dean. The instructor and the student will confer regularly regarding the student's progress.
A student may contract for one to six credit hours of independent study through an arrangement with an instructor who agrees to direct such a study. The student will submit a plan acceptable to the instructor and to the department chair. The instructor and student will confer regularly regarding the process of the study. |
Glen Ridge, FL PrecalculusThe course also gives the students a thorough introduction of functions and quadratic equations. They will learn all the operations associated with the two, such as factoring and graphing. The end of the course takes the students?Precalculus includes algebra, coordinate geometry, trigonometry, and most of all, functions---the general concept as well as specific functions. Students come to this course familiar with basic algebra and geometry. A precalculus course builds on all of their previous mathematical knowledge and experience to understand calculus that is essential to succeed in a college |
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Show More equations that describe the underlying physical properties. However, these equations are usually much too complicated to solve, either by the smartest mathematician or the largest supercomputer. This problem is overcome by constructing an approximation: a numerical model with a simpler structure can be translated into a program that tells the computer how to carry out the simulation.This book conveys the fundamentals of mathematical models, numerical methods and algorithms. Opening with a tutorial on mathematical models and analysis, it proceeds to introduce the most important classes of numerical methods, with finite element, finite difference and spectral methods as central tools. The concluding section describes applications in physics and engineering, including wave propagation, heat conduction and fluid dynamics. Also covered are the principles of computers and programming, including MATLAB® |
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