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Elementary Algebra - 8th edition
Algebra is accessible and engaging with this popular text from Charles ?Pat? McKeague! ELEMENTARY ALGEBRA is infused with McKeague?s passion for teaching mathematics. With years of classroom experience, he knows how to write in a way that you will understand and appreciate. McKeague?s attention to detail and exceptionally clear writing style help yo...show moreu to move through each new concept with ease. Real-world applications in every chapter of this user-friendly book highlight the relevance of what you are learning. And studying is easier than ever with the book?s multimedia learning resources, including ThomsonNOW for ELEMENTARY ALGEBRA, a personalized online learning companion00 +$3.99 s/h
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Textbookcenter.com Columbia, MO
Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book
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bluehouse acton, MA
Brand new.
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Lyric Vibes Geneva, IL
Hardcover New 0495108391 New Condition ~~~ Right off the Shelf-BUY NOW & INCREASE IN KNOWLEDGE...
$229
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hematics: Its Power and Utility
Explores the power and historic impact of mathematics and helps students harness that power by developing an effective approach to problem solving. ...Show synopsisExplores the power and historic impact of mathematics and helps students harness that power by developing an effective approach to problem solving. This title also explores the utility and application of math concepts to a wide variety of real-life situations: money management; handling of credit cards; inflation; and, many other topics.Hide synopsis
Description:Very good. Hardcover. Instructor Edition: Same as student...Very good. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Has minor wear and/or markings. SKU: 9781111581527-3Fine. Hardcover. Instructor Edition: Same as student edition...Fine. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Almost new condition. SKU: 9781111581527-2New. 1111577420 #Instructor's Edition. Identical to student...New. 1111577420 #Instructor's Edition. Identical to student edition except has publisher notations on cover and extra information for professors. Great way to save on this book. WE SHIP DAILY! !
Description:Good. This is an INSTRUCTOR COPY. INSTRUCTOR COPY. May have...Good. This is an INSTRUCTOR COPY. INSTRUCTOR COPY. May have minimal notes/highlighting, minimal wear/tear. Please contact us if you have any Questions.
Reviews of Mathematics: Its Power and Utility
Excellent material for pre-high school studants, or for people who want to advance their understanding of math. Includes Alegebraic problem solving, Geometry,Statistics, Logic and Sets.
Well written, lots of illustrations
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guide updated and expanded for today's mathphobes Written by two pioneers of the concept of math anxiety and how to overcome it, Arithmetic and Algebra Again has helped tens of thousands of people conquer their irrational fear of math. This revised and expanded second edition of the perennial bestseller: Features the latest techniques for breaking through common anxieties about numbers Takes a real-world approach that lets mathphobes learn the math they need as they need it Covers all key math areas--from whole numbers and fractions to basic algebra Features a section on practical math for banking, mortgages, interest, and statistics and probability Includes a new section on the graphing calculator, a chapter on the metric system, a section on word problems, and all updated exercises
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Loci Browse Articles
The purpose of this collection of applets and activities is to make students familiar with the basic principles of complex numbers. Combining explanatory text, exercises and interactive GeoGebra applets, this resource is suitable for both classroom lectures and distance learning.
Geometry Playground (v1.3) is a free Java application for doing "ruler and compass" constructions in both Euclidean, Spherical, Projective, Hyperbolic, Toroidal, Manhattan and Conical geometries. Its purpose is to help users develop a familiarity with various conceptualizations of these geometries.
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O Level Mathematics (syllabus D) (4024)O Level Mathematics (syllabus D) (4024) 1/1 O Level Mathematics (syllabus D) (4024) Will students be given a formula sheet to help them in the exam or do they need to
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0321127110
9780321127112
Beginning Algebra:The Lial series has helped thousands of students succeed in developmental mathematics through its friendly writing style, numerous realistic examples, extensive problem sets, and complete supplements package. In keeping with its proven track record, this revision includes a new open design, more exercises and applications, and additional features to help both students and instructors succeed.
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Rent Beginning Algebra 9th edition today, or search our site for Margaret L. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
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Free Online Calculus Courses
Calculus Content Navigation
Building on principles learned in algebra and geometry, calculus deals with limits. Two sub-categories of calculus are differential calculus and integral calculus. Integral calculus deals with the idea of accumulation, while differential calculus deals with the rate of change. These calculus courses and lectures give you insight into this branch of mathematics and help you understand the concepts involved.
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Lesson study is a professional development process that teachers engage in to systematically examine their practice, with the goal of becoming more effective. Originating in Japan, lesson study has gained significant momentum in the mathematics education community in recent years. As a process for professional development, lesson study became highly... more...
Build student success in math with the only comprehensive guide for developing math talent among advanced learners. The authors, nationally recognized math education experts, offer a focused look at educating gifted and talented students for success in math. More than just a guidebook for educators, this book offers a comprehensive approach to mathematics... more...
ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing... more...
This Ebook is designed for science and engineering students taking a course in numerical methods of differential equations. Most of the material in this Ebook has its origin based on lecture courses given to advanced and early postgraduate students. This Ebook covers linear difference equations, linear multistep methods, Runge Kutta methods and finite... more...
This book covers the discourse and equity in mathematics education research. Given the inherent connection between discourse and equity, this book focuses on two approaches to the connection. Contributors consider the ways in which the social, mathematical, cultural, and political aspects of classroom interactions impact students' opportunities... more...
This book considers the views of participants in the process of becoming a mathematician, that is, the students and the graduates. This book investigates the people who carry out mathematics rather than the topics of mathematics. Learning is about change in a person, the development of an identity and ways of interacting with the world. It investigates... more...
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Using spreadsheets to foster algebraic reasoning in the middle school mathematics classroom
Author:
Chávez-López, Óscar, 1963-
Date:
2003-04-11
Abstract:
What is Algebra? For most people, algebra means using and manipulating (algebraic)
symbols, and solving equations. The Algebra Standard, as it is stated in the Principles and Standards for School Mathematics, expects more than that from students and teachers. In this workshop we will address some of the ways in which spreadsheets can help to promote algebraic thinking in the middle school mathematics classroom.
URI:
Items in MOspace are protected by copyright, with all rights reserved, unless otherwise indicated.
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Summary: Here's the perfect self-teaching guide to help anyone master differential equations--a common stumbling block for students looking to progress to advanced topics in both science and math. Covers First Order Equations, Second Order Equations and Higher, Properties, Solutions, Series Solutions, Fourier Series and Orthogonal Systems, Partial Differential Equations and Boundary Value Problems, Numerical Techniques, and more.
A. Vector Fields B. Flows and Trajectories C. Poincare-Bendixon Theory V. Power Series Solutions A. Review of Key Properties of Power Series B. Series Solutions for First Order Equations C. Second Order Linear: Ordinary Points D. Regular Singular Points1440259
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GED Mathematics review Product Description
This overview of high school mathematics--including arithmetic, charts and graphs, probability, statistics, algebra, geometry, number operations, data analysis, and coordinate geometry--is designed to aid viewers in passing the GED test. GED Mathematics movie Practice problems, test-taking strategies, and time saving tips are also featured as viewers are guided by a friendly math professor at the chalkboard, offering step-by-step solutions and clear explanations along the way. ...See Full Description
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Students can put fractions into a real-life context by using pizza. Developmental math instructors can use these examples to explain more complex mathematical concepts, or create new activities and homework assignments...
Giorgio Ingargiola, Associate Profession of Computer and Information Science at Temple University, has created the Wumpus World as an example of knowledge representation, reasoning, and planning to "introduce the...
This page, presented by the College Board, describes the course and exams for high school Advanced Placement Calculus AB focusing on differential and integral calculus. The three broad topics here are: Functions,...
This page, presented by the College Board, describes the course and exams for high school Advanced Placement Calculus BC focusing on differential and integral calculus (covered in Calculus AB) as well as additional...
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integralCALCCourse Description
Need some tips for Calculus 1A? Or maybe you're madly reviewing for tomorrow's math test? Either way, never fear - Krista, an experienced math tutor, will help you understand the world of calculus, step-by-step. Start by learning the difference between a function and an equation - and how to analyze a function's graph for continuity and limits. Then, step into the world of tangent lines, differentiation, and more. Each lesson includes examples and sample problems to help you along the way.
Lessons in this Course
1.
Functions vs. Equations
5:22
2.
How to Use the Vertical Line Test
2:58
3.
Limits and Continuity
6:26
4.
Prove the Limit Doesn't Exist | Example
5:37
5.
Precise Definition of a Limit | Example
9:19
6.
Derivatives
6:53
7.
Definition of the Derivative | Example
3:43
8.
Equation of the Tangent Line | Example
16:49
9.
Implicit Differentiation | Example
9:35
10.
Optimization
8:55
11.
Related Rates
8:16
What is included in the course?
All of the video-based lessons listed on the Course Description tab, including interactive exercises and attached
files you can use along with the lesson. You also can ask the teacher (and other students) questions, and submit a
video or photo of your work to get direct feedback from the teacher.
What is Curious.com?
Curious.com is a site that enables teachers like integralCALC to make money by
teaching online to students around the world.
Where does my money go?
Most of the money goes directly to the teacher. The rest goes to Curious for the hardware and software and human
support required to make the delivery of this awesome course possible.
How do I access the course when I want to learn?
You can access the course, and any other Curious lessons you have enrolled in, by logging into on
your computer or tablet. You will be prompted to create an account when you purchase the course if you don't have one
already.
How long do I have access to the course?
For life. Really.
What if I don't learn, or don't like it?
We are confident you will love this course--like literally thousands of others before you--but if you don't for any
reason we will be happy to refund your money and disable your access to it.
vanessa v comment: whats the easiest way to learn calculus when you're 10 years old?
Krista K comment: make sure first that you're really good with algebra. from there, google "calculus and limits".... and let me know if you have trouble! :)
Allan P comment: Thanks for the refresher. I think this course will be very useful. One Love!
joni d comment: It's been 40 years since I took calculus. This is on my bucket list!
bailey m comment: when I was in the sixth grade i didn't turn my homework in so they stuck me ion regular math for another year so in the eighth grade i took pre-algebra now I'm getting ready to go into the ninth grade and i really want to get a good grade in algebra. How would I go about doing that? do you have any suggestions?
Shane B comment: Found it a bit complicated. haven't done calculas since high school in the 70s'.
Christopher T comment:
"jacinta k commented:
If i invest K50 a year for 40 years toward my POSF savings, and earn 8% a year on my investments, how much will i have when i retire?"
Answer: 1,086,226.075
Students in this lesson (567)
About the TeacherTable of Contents
1.
Lesson Intro
0:23
2.
Functions vs. Equations
0:23
3.
What are Functions?
1:06
4.
Domain and Range
1:32
5.
What Functions are Not
1:02
6.
Combinations and Compositions
0:54
Lesson Description
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Practice Makes Perfect Precalculus - 12 edition
Summary: Don't be perplexed by precalculus. Master this math with practice, practice, practice! Practice Makes Perfect: Precalculus is a comprehensive guide and workbook that covers all the basics of precalculus that you need to understand this subject. Each chapter focuses on one major topic, with thorough explanations and many illustrative examples, so you can learn at your own pace and really absorb the information. You get to apply your knowledge and practice what you've learned through...show more a variety of exercises, with an answer key for instant feedback. Offering a winning solution for getting a handle on math right away, Practice Makes Perfect: Precalculus is your ultimate resource for building a solid understanding of precalculus fundamentals. ...show less
Textbook may contain underlining, highlighting or writing. Infotrac or untested CD may not be included.
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Chapter 3: Geometry TE - Enrichment
The goal of an enrichment section is just what is implied in the title, "to enrich." By enrichment, we mean something that breathes a new or different life into something else- to make it better to enliven it. This is the goal of this branch of the teacher's edition. This is an opportunity for you and your students to locate and explore the wonderful world of geometry in other subjects such as architecture or music or art. It is a chance for students to see how the world of mathematics can connect to other subjects that they are passionate about.
Our goal is that using this Enrichment Flexbook will help you to expand your own personal creativity as well as the creativity of your students. The projects/topics in this flexbook can be used in several different ways. They can be used as a discussion point, an example to highlight during a lesson, a project to expand on whether students complete the project in class or at home or as a way to broaden student thinking by using a web search once per week as an example. It is not the intention that every single lesson be used in this flexbook. Take what inspires you and use it to inspire your students. Isn't that what the world of mathematics is all about!
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...
Show More with: Exercises in: Hilbert space theory, Lie groups, Matrix-valued differential forms, Bose-Fermi operators and string theory. All other chapters have been updated with new problems and materials. Most chapters contain an introduction to the subject discussed in the text. Complex Numbers and Functions Sums and Products Discrete Fourier Transform Algebraic and Transcendental Equations Vector and Matrix Calculations Matrices and Groups Matrices and Eigenvalue Problems Functions of Matrices Transformations L'Hospital's Rule Lagrange Multiplier Method Linear Difference Equations Linear Differential Equations Integration Continuous Fourier Transform Complex Analysis Special Functions Inequalities Functional Analysis Combinatorics Convex Sets and Functions Optimization
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More About
This Textbook
Overview
The Companion Guide to The Mathematical Experience, Study Edition has been created as a teaching tool, not only for the teacher and the student, but also for those students who are potential teachers. Its major purpose is to enhance the value of The Mathematical Experience, Study Edition as a textbook for teachers and to provide content and method for prospective teachers. Thus, unlike instructional guides that are available to the adopting teacher only, this Companion is available to the student or the teacher who wants independently to develop further skills in teaching mathematics. An additional value is that it provides suggested topics to explore that are not in the text but that coordinate beautifully to the text. The inclusion of these topics makes The Companion Guide a flexible teaching tool, adaptable to a variety of courses and useable with many individual selections of other course materials. The Companion Guide is rich in suggestions for classroom discussion topics. Each is linked to a chapter of the textbook and to the central idea of learning how to think, talk, and write ABOUT mathematics while learning how to DO mathematics. It provides insights into the subtleties of mathematical concepts and warns of pitfalls where ambiguity and misunderstanding often arise. It is a wealth of experience with ideas that WORK, gained through live classroom interaction by the authors and shared in this book with
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More About
This Textbook
Overview
Explains how to reason and model combinatorially. Enables students to develop proficiency in fundamental discrete math problem solving in the manner that a calculus textbook develops competence in basic analysis problem solving. Stresses the systematic analysis of different possibilities, exploration of the logical structure of a problem and ingenuity. This edition contains many new exercises.
Editorial Reviews
From The Critics
This text for undergraduate and beginning graduate students in computer science and mathematics covers the theory and application of combinatorial reasoning. Tucker (mathematics, Stony Brook) begins with a discussion of the elements of graph theory before moving on to enumeration. The systematic analysis of different possibilities, the exploration of the logical structure of a problem, and ingenuity are emphasized throughout the text. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
"...a well-structured text that addresses a broad range of topics... It is well presented, written clearly and easy to follow." (Times Higher Education Supplement, November
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MPM1D Principles of Mathematics - Course Outline
Mr. Krūmins Email: kruminsa@hdsb.ca Website:
Office Hours: first 15 minutes at Room: 224
the start of lunch
This course enables students to develop an understanding of mathematical concepts related to algebra, analytic geometry, and
measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate
relationships, which they will then generalize as equations of lines, and will determine the connections between different
representations of a linear relation. They will also explore relationships that emerge from the measurement of three-dimensional
figures and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multi-step
problems.
Curriculum
A student's final report card grade will be based on the evidence provided of these overall curriculum expectations:
Process Expectations
Students will be actively engaged in the following seven processes which are integrated into all areas of the course: problem
solving, reasoning and proving, reflecting, connecting, representing, selecting tools and computational strategies and
communicating.
Number Sense and Algebra
• demonstrate an understanding of the exponent rules of multiplication and division, and apply them to simplify expressions;
• manipulate numerical and polynomial expressions, and solve first-degree equations.
Linear Relations
• apply data-management techniques to investigate relationships between two variables;
• demonstrate an understanding of the characteristics of a linear relation;
• connect various representations of a linear relation.
Analytic Geometry
• determine the relationship between the form of an equation and the shape of its graph with respect to linearity and non-linearity;
• determine, through investigation, the properties of the slope and y-intercept of a linear relation;
• solve problems involving linear relations.
Measurement and Geometry
• determine, through investigation, the optimal values of various measurements;
• solve problems involving the measurements of two-dimensional shapes and the surface areas and volumes of three-dimensional
figures;
• verify, through investigation facilitated by dynamic geometry software, geometric properties and relationships involving two-
dimensional shapes, and apply the results to solving problems..
1
Your Report Card Grade will be determined as follows:
Term work: 25% Knowledge & Understanding: Knowledge of content and the understanding of
70% of your grade will be mathematical concepts.
based on all of the evidence
15% Application: the application of knowledge and skills in familiar contexts; transfer of
you have provided. It will
knowledge and skills to new contexts; making connections within and between various contexts.
reflect your most consistent
level of achievement with 20% Thinking: use of planning and processing skills; use of critical and creative thinking
special consideration given to processes.
more recent evidence.
10% Communication: Expression and organization of ideas and mathematical thinking,
communication for different audiences/purposes and use of conventions, vocabulary and
terminology of the discipline … all using oral, visual and written forms.
Final Evaluation: 15% Performance Task: Consisting of a mathematical investigation or contextual, open-ended
30% of your grade will be problematic situation suited to a variety of approaches including use of technology where
determined at the end of the appropriate.
course. 15% Exam: Consisting of a variety of question types (e.g. short answer, multiple choice,
extended tasks) sampling all strands and categories of 2.5 hours duration or less.
Your final grade will be calculated by combining your Term (70%) grade and your Exam and Performance Task Evaluations
(30%).
Academic Standards
It is your responsibility to provide evidence of your learning within established timelines. Due dates for assignments and
the scheduling of tests will be communicated well in advance to allow you to schedule your time. If you aren't going to
be able to follow an agreed upon timeline you should demonstrate your responsibility and organizational skills by
discussing with your teacher the challenges you're facing as far in advance of the deadline as possible.
It is your responsibility to be academically honest in all aspects of your schoolwork so that the marks you receive are a
true reflection of your achievement.
Plagiarism is using the words, ideas or work of someone else without giving appropriate credit to the original creator.
This is a form of cheating.
Consequences for not meeting these academic standards may include:
Reporting the issue to your parents;
Requiring you to complete the original or alternative work after school or during your lunch hour;
Requiring you to complete an alternative assignment;
Suspension;
Assigning a "zero" for an assignment not completed prior to an agreed upon closure date;
Mark deduction of 5% / day.
NOTE: the complete HDSB policies and administrative procedures for "Lates and Missed Assignments" and "Cheating
and Plagiarism" policies may be found at
2
Learning Skills & Work Habits
These learning skills and work habits will be taught, assessed and evaluated throughout the course.
3
4
Unit Outlines
Curriculum
Units Major Assignments / Evaluations Key Resources
Focus
1 BEDMAS; Unit/Diagnostic Test Handouts
Numeracy Integers; Quizzes Textbook
Fractions; TIPS Assignment
Percents;
Exponents with
numeric base
2 Operations with Unit Test Handouts
polynomials
Algebra and Quizzes Textbook
Polynomials Exponent Laws
TIPS Assignment
3 Solving and Unit Test Handouts
checking
Equations equations; Quizzes Textbook
solving word
problems TIPS Assignment
4 Surface area Unit Test Handouts
and volume of
Measurement polyhedra; Quizzes Textbook
Pythagorean TIPS Assignment Manipulatives
Theorem
Computers
5 Scatter plots; Unit Test Handouts
line of best fit,
Relationships graphing linear Quizzes Textbook
and non-linear
relationships; TIPS Assignment Graphing Calculators
CBR's
Computers
6 Slope of a line; Unit Test Handouts
direct and
Slope and the partial variation Quizzes Textbook
Line
y=mx+b; TIPS Assignment Graphing Calculators
ax+by+c=0;
applications of Computers
linear relations;
finding
equations of
5
lines
7 Angles in Unit Test Handouts
triangles;
Geometry properties of Quizzes Textbook
quadrilaterals
TIPS Assignment Computers
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On the Home screen, you can enter mathematical expressions and functions, along with other instructions. The answers are displayed on the Home screen. The TI-36X Pro screen can display a maximum of four lines with a maximum of 16 characters per line. For entries and expressions of more than 16 characters, you can scroll left and right (!and ") to...
When you calculate an entry on the Home screen, depending upon space, the answer is displayed either directly to the right of the entry or on the right side of the next line. Special indicators and cursors may display on the screen to provide additional information concerning functions or results.
Indicator Definition MathPrint™ cursor. Continue entering the current MathPrint™ element, or press an arrow key to exit the element. 2nd functions Most keys can perform more than one function. The primary function is indicated on the key and the secondary function is displayed above it.
SCI expresses numbers with one digit to the left of the decimal and the appropriate power of 10, as in 1.2345678 5 (which is the same as 1.2345678×10 ENG displays results as a number from 1 to 999 times 10 to an integer power.
Multi-tap keys A multi-tap key is one that cycles through multiple functions when you press it. For example, the X key contains the trigonometry functions sin and sin as well as the hyperbolic functions sinh and sinh . Press the key repeatedly to display the function that you want to enter.
Answer toggle Press the r key to toggle the display result (when possible) between fraction and decimal answers, exact square root and decimal, and exact pi and decimal. Pressing r displays the last result in the full precision of its stored value, which may not match the rounded value.
3 % c % i < Order of operations The TI-36X Pro calculator uses Equation Operating System (EOS™) to evaluate expressions. Within a priority level, EOS evaluates functions from left to right and in the following order. Expressions inside parentheses.
Clearing and correcting Returns to the Home screen. Clears an error message. Clears characters on entry line. Moves the cursor to last entry in history once display is clear. Deletes the character at the cursor. Inserts a character at the cursor. Clears variables x, y, z, t, a, b, c, and d to their default value of 0.
•...
³ Problem A mining company extracts 5000 tons of ore with a concentration of metal of 3% and 7300 tons with a concentration of 2.3%. On the basis of these two extraction figures, what is the total quantity of metal obtained? If one ton of metal is worth 280 dollars, what is the total value of the metal extracted? 3 % _ V 5000 <...
Powers, roots and inverses Calculates the square of a value. The TI-36X Pro calculator evaluates expressions entered with F and a from left to right in both Classic and MathPrint™ modes. Raises a value to the power indicated. Use "...
Number functions d NUM d " displays the NUM menu: 1: abs( Absolute value 2: round( Rounded value 3: iPart( Integer part of a number 4: fPart( Fractional part of a number 5: int( Greatest integer that is the number 6: min( Minimum of two numbers 7: max(...
90 U % i < % b 3 F T 7 F < To one decimal place, the measure of angle A is 66.8¡, the measure of angle B is 23.2¡, and the length of the hypotenuse is 7.6 meters. Hyperbolics Z (multi-tap keys) Pressing one of these multi-tap keys repeatedly lets you...
Logarithm and exponential functions C (multi-tap keys) D yields the logarithm of a number to the base e (e ≈ 2.718281828459). D D yields the common logarithm of a number. C raises e to the power you specify. C C raises 10 to the power you specify. Examples D D1 ) <...
q $$ """" < z G 3 " U 4 z "" 2 P % b 3 < ------ - The slope of the tangent line at x = is zero. A maximum or minimum of the function must be at this point! Numeric integral % Q calculates the numeric function integral of an expression with respect to a variable x, given a lower limit and...
< Notice that both areas are equal. Since this is a parabola with the vertex at (4,0) and zeros at (M2, 0) and (2, 0) you see that the symmetric areas are equal. Stored operations % n lets you store a sequence of operations. % m plays back the operation.
z
L z z < < W 4 < ³ Problem In a gravel quarry, two new excavations have been opened. The first one measures 350 meters by 560 meters, the second one measures 340 meters by 610 meters. What volume of gravel does the company need to extract from each excavation to reach a depth of 150 meters? To reach 210 meters? Display the results in engineering notation.
210 V % h < < 150 V z z < 210 V z z < For the first excavation: The company needs to extract 29.4 million cubic meters to reach a depth of 150 meters, and to extract 41.16 million cubic meters to reach a depth of 210 meters.
< v < % ˜ < Notice L2 is calculated using the formula you entered, and L2(1)= in the author line is highlighted to indicate the list is the result of a formula. ³ Problem On a November day, a weather report on the Internet listed the following temperatures.
5: Binomcdf Computes a cumulative probability at x for the discrete binomial distribution with the specified numtrials and probability of success (p) on each trial. x can be non- negative integer and can be entered with options of SINGLE, LIST or ALL (a list of cumulative probabilities is returned.) 0 { p { 1 must be true.
sx or sy Population standard deviation of x or y. Gx or Gy Sum of all x or y values. or Gy Sum of all x or y values. Sum of (x…y) for all xy pairs. a (2-Var) Linear regression slope. b (2-Var) Linear regression y-intercept.
This line of best fit, y'=0.67732519x'N18.66637321 models the linear trend of the data. Press $ until y' is highlighted. < 55 ) < The linear model gives an estimated braking distance of 18.59 meters for a vehicle traveling at 55 kph. Regression example 1 Calculate an ax+b linear regression for the following data: {1,2,3,4,5};...
< 0 < 1 < << Warning: If you now calculate 2-Var Stats on your data, the variables a and b (along with r and r ) will be calculated as a linear regression. Do not recalculate 2-Var Stats after any other regression calculation if you want to preserve your regression coefficients (a, b, c, d) and r values for your particular problem in the StatVars menu.
< Probability % † H is a multi-tap key that cycles through the following options: A factorial is the product of the positive integers from 1 to n. n must be a positive whole number { 69. Calculates the number of possible combinations of n items taken r at a time, given n and r.
2: Edit function Lets you define the function f(x) and generates a table of values. The function table allows you to display a defined function in a tabular form. To set up a function table: 1. Press I and select Edit function. 2.
< After searching close to x = 18, the point (18, 324) appears to be the vertex of the parabola since it appears to be the turning point of the set of points of this function. To search closer to x = 18, change the Step value to smaller and smaller values to see points closer to (18, 324).
< rref % t " # < % t <) < Notice that [A] has an inverse and that [A] is equivalent to the identity matrix. Vectors In addition to those in the Vector MATH menu, the following vector operations are allowed. Dimensions must be correct: •...
% … MATH % … " displays the vector MATH menu, which lets you perform the following vector calculations: 1: DotProduct Syntax: DotP(vector1, vector2) Both vectors must be the same dimension. 2: CrossProduct Syntax: CrossP(vector1, vector2) Both vectors must be the same dimension.
Example of quadratic equation Reminder: If you have already defined variables, the solver will assume those values. Poly-solv % Š Enter < coefficients < Solutions < Note: If you choose to store the polynomial to f(x), you can use I to study the table of values.
Constants Constants lets you access scientific constants to paste in various areas of the TI-36X Pro calculator. Press % Πto access, and ! oro" to select either the NAMES or UNITS menus of the same 20 physical constants.Use # and $ to scroll through the list of constants in the two menus.
Note: Displayed constant values are rounded. The values used for calculations are given in the following table. Constant Value used for calculations speed of light 299792458 meters per second gravitational 9.80665 meters per second acceleration Planck's constant 6.62606896×10 Joule seconds NA Avogadro's number 6.02214179×10 molecules per mole...
Conversions The CONVERSIONS menu permits you to perform a total of 20 conversions (or 40 if converting both ways). To access the CONVERSIONS menu, press % –. Press one of the numbers (1-5) to select, or press # and $ to scroll through and select one of the CONVERSIONS submenus.
• You press < on a blank equation or an equation with only numbers. Invalid Data Type — In an editor, you entered a type that is not allowed, such as a complex number, matrix, or vector, as an element in the stat list editor, matrix editor and vector editor.
STAT — You attempted to calculate 1-var or 2-var stats with no defined data points, or attempted to calculate 2-var stats when the data lists are not of equal length. SYNTAX — The command contains a syntax error: entering more than 23 pending operations or 8 pending values; or having misplaced functions, arguments, parentheses, or commas.
Discard used batteries according to local regulations. How to remove or replace the battery The TI-36X Pro calculator uses one 3 volt CR2032 lithium battery. Remove the protective cover and turn the calculator face downwards.
Dispose of the dead battery immediately and in accordance with local regulations. Per CA Regulation 22 CCR 67384.4, the following applies to the button cell battery in this unit: Perchlorate Material - Special handling may apply. See In case of difficulty Review instructions to be certain calculations were performed properly.
Customers in the U.S., Canada, Mexico, Puerto Rico and Virgin Islands: Always contact Texas Instruments Customer Support before returning a product for service. All other customers: Refer to the leaflet enclosed with this product (hardware) or contact your local Texas Instruments retailer/distributor.
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Math Survival Guide Tips and Tricks for Science Students
9780471270546
ISBN:
0471270547
Edition: 2 Pub Date: 2003 Publisher: Wiley & Sons, Incorporated, John
Summary: This second edition of 'Math Survival Guide' provides tips for science students in the form of a quick reference/update guide. It uses an approachable tone and appropriate level and includes good problem sets.
Appling, Jeffrey R. is the author of Math Survival Guide Tips and Tricks for Science Students, published 2003 under ISBN 9780471270546 and 0471270547. Five hundred sixty eight Math Survival Guide Tips ...and Tricks for Science Students textbooks are available for sale on ValoreBooks.com, one hundred fifty three used from the cheapest price of $7.69, or buy new starting at $32.36.[read more [more will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions.[less]
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Math
A necessary life skill and more, Math Homeschool Curriculum from Mardel helps homeschool teachers take their student from basic addition and subtraction to algebra to geometry and even pre-calculus and beyond. With curriculum from noteworthy and acclaimed publishers including ASCI, BJU Press, Saxon and many more, homeschool teachers are certain to find the method and curriculum that works best for their homeschool student. Homeschool math curriculum includes workbooks and texts for students as well as teacher lesson plans and other resources. Find a great selection of Math Homeschool Curriculum at Mardel.
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Summary: As in previous editions, the focus in ALGEBRA: INTRODUCTORY & INTERMEDIATE remains on the Aufmann Interactive Method (AIM). Users are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. The role of ''active participant'' is crucial to success. Presenting students with worked examples, and then providing them with the opportunity to immediately work similar problems, he...show morelps them build their confidence and eventually master the3.083.26 +$3.99 s/h
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Teachers!Prepare...Math.F/Sat*1 - 96 edition
ISBN13:978-0803964815 ISBN10: 0803964811 This edition has also been released as: ISBN13: 978-0803964167 ISBN10: 0803964161
Summary: If you're a teacher or administrator who wants to give your students the best chances possible on the Math SAT, or if you're a parent of a student facing the test, here is the advantage you're looking for. Designed as a companion book to Students! Get Ready for the Mathematics for SAT I, this workbook is crammed with strategies and ideas. Show your students techniques that can make a big difference in their college chances--teach them techniques that will really boost their Math SAT ...show moreI scores. This information-packed teacher's volume gives you: * An overview of the SAT I, including a description of its format, content, and the use of a calculator on the test * A selective review of the mathematics taught through elementary algebra and geometry, with particular attention to problem solving. * Less-well-known mathematics ''facts'' and problem-solving tools * Ways to advise students on strategies for taking the SAT I, including when and how to guess on unfamiliar items * A detailed presentation of specific problem-solving strategies Follow the step-by-step plan in this book, and you will help to signicantly increase your students' scores. Hand your students these tools and they'll be prepared to face the SAT I
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Odenton PrealMATLAB is used in the course to some extent. MATLAB stands for Matrix Laboratory and involves the formulation of a problem in matrix terms. Matlab can handle vast amounts of input data and manipulate the data in accordance with the instructions that the user provides.
...Chemistry is the study of matter, its composition, properties and methods of producing change. More than any other branch of mathematics, Geometry deals with logic and reasoning and its application in problem solving. What is the SAT?
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This course provides an introduction to SciPy. It is intended to
serve as an introduction to numerical programming in Python with
SciPy for those who are new to the use of numerical tools for
Python. A mathematical background would be helpful and, in
particular, will help the student to get more benefit from the
course. But, it is expected that the student will still benefit
from the course, with or without that background.
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Improving your child's knowledge of trigonometry is easy to do with Lifepac Pre-Calculus Unit 7 Worktext! The colorful, print-based worktext from Alpha Omega Publications will help your high school student learn inverse functions, convert polar coordinates, as well as how to graph polar equations. Tests are included. Format: Paperback. Grade Level: 12th Grade.
Want to help your high school student improve his advanced math skills? Then teach him with the Lifepac Pre-Calculus Unit 5 Worktext! The consumable, print-based worktext includes lessons on important math topics like identities and functions, Pythagorean relations, and trigonometric identities. Tests are included. Format: Paperback. Grade Level: 12th Grade.
The Saxon Math program is probably the biggest news in Math in our generation. It has turned math-hating children into children whose favorite subject is Math! Children who have worked with this program have exhibited great gains on standardized tests. The secret is in presentation. Mary Pride's Reader Award winner!
Contains over 100 hours of Advanced Math content, including instruction for every part of every lesson, as well as complete solutions for every example problem, practice problem, problem set, and test problem.
The user-friendly CD format offers students helpful navigation tools within a customized player and is compatible with both Windows and Mac
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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QS099 Modern Elementary College Algebra. This course is an introductory course presenting the principles of elementary algebra. Topics covered include the real number system, linear equations and inequalities, factoring, operations with polynomials, exponents and radicals, and an introduction to functions and the Cartesian coordinate system. Placement into this course is done through the placement testing program. (3 credits)
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College Algebra and Trigonometry: A Unit Circle Approach (5th Edition)
Book Description: Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Fifth Edition gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find enoughcarefully placed learning aids and review tools to help them do the math without getting distracted from their objectives. Regardless of their goals beyond the course, all students will benefit from Dugopolski's emphasis on problem solving and critical thinking, which is enhanced by the addition of nearly 1,000 exercises in this edition
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Thanks for visiting ARIS or MathZone. We have retired ARIS and MathZone, but no worries! We've replaced them with Connect and ConnectPlus, our new generation of digital learning products with improved user experience and enhanced functionality.
Algebra for College Students, 5e is part of the latest offerings in the successful Dugopolski series in mathematics. The author's goal is to explain mathematical concepts to students in a language they can understand. In this book, students and faculty will find short, precise explanations of terms and concepts written in understandable language. The author uses concrete analogies to relate math to everyday experiences. For example, when the author introduces the Commutative Property of Addition, he uses a concrete analogy that "the price of a hamburger plus a Coke is the same as a Coke plus a hamburger". Given the importance of examples within a math book, the author has paid close attention to the most important details for solving the given topic. Dugopolski includes a double cross-referencing system between the examples and exercise sets, so no matter which one the students start with, they will see the connection to the other. Finally, the author finds it important to not only provide quality, but also a good quantity of exercises and applications. The Dugopolski series is known for providing students and faculty with the most quantity and quality of exercises as compared to any other developmental math series on the market. In completing this revision, Dugopolski feels he has developed the clearest and most concise developmental math series on the market, and he has done so without comprising the essential information every student needs to become successful in future mathematics courses. The book is accompanied by numerous useful supplements, including McGraw-Hill's online homework management system, MathZone.
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Port Republic, MD CalculusMatlab can handle vast amounts of input data and manipulate the data in accordance with the instructions that the user provides. It has amazing plotting capabilities with both 2-D and 3-D plots. It also provides a vast array of statistical functions including means, variances, medians, and modes of data sets.
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Elementary Classical Analysis courses in advanced calculus and introductory real analysis,Elementary Classical Analysisstrikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.
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... level, while maintaining an overall primary goal: to have math make sense from ... Make and explain predications based on data. 232A-232B, 232-235 D77 Make predictions to answer ...
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This module is included inLens:Community College Open Textbook Collaborative By: CC Open Textbook CollaborativeAs a part of collection: "Elementary Algebra"
Comments:
"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"
Click the
"College Open Textbooks"
link to see all content they endorseBasic Properties of Real Numbers: The Real Number Line and the Real Numbers
Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.
Objectives of this module: be familiar with the real number line and the real numbers, understand the ordering of the real numbers.
Overview
The Real Number Line
The Real Numbers
Ordering the Real Numbers
The Real Number Line
Real Number Line
In our study of algebra, we will use several collections of numbers. The real number line allows us to visually display the numbers in which we are interested.
A line is composed of infinitely many points. To each point we can associate a unique number, and with each number we can associate a particular point.
Coordinate
The number associated with a point on the number line is called the coordinate of the point.
Graph
The point on a line that is associated with a particular number is called the graph of that number.
We construct the real number line as follows:
Construction of the Real Number Line
Draw a horizontal line.
Choose any point on the line and label it 0. This point is called the origin.
Choose a convenient length. This length is called "1 unit." Starting at 0, mark this length off in both directions, being careful to have the lengths look like they are about the same.
We now define a real number.
Real Number
A real number is any number that is the coordinate of a point on the real number line.
Positive and Negative Real Numbers
The collection of these infinitely many numbers is called the collection of real numbers. The real numbers whose graphs are to the right of 0 are called the positive real numbers. The real numbers whose graphs appear to the left of 0 are called the negative real numbers.
The number 0 is neither positive nor negative.
The Real Numbers
The collection of real numbers has many subcollections. The subcollections that are of most interest to us are listed below along with their notations and graphs.
Natural Numbers
The natural numbers (N)(N):{1,2,3,…}{1,2,3,…}
Whole Numbers
The whole numbers (W)(W):{0,1,2,3,…}{0,1,2,3,…} Notice that every natural number is a whole number.
Integers
The integers (Z)(Z):{…,−3,−2,−1,0,1,2,3,…}{…,−3,−2,−1,0,1,2,3,…} Notice that every whole number is an integer.
Rational Numbers
The rational numbers (Q)(Q): Rational numbers are real numbers that can be written in the form a/ba/b, where aa and bb are integers, and b≠0b≠0.
Fractions
Rational numbers are commonly called fractions.
Division by 1
Since bb can equal 1, every integer is a rational number: a1=aa1=a.
Division by 0
Recall that 10/2=510/2=5 since 2⋅5=102⋅5=10. However, if 10/0=x10/0=x, then 0⋅x=100⋅x=10. But 0⋅x=00⋅x=0, not 10. This suggests that no quotient exists.
Now consider 0/0=x0/0=x. If 0/0=x0/0=x, then 0⋅x=00⋅x=0. But this means that xx could be any number, that is, 0/0=40/0=4 since 0⋅4=00⋅4=0, or 0/0=280/0=28 since 0⋅28=00⋅28=0. This suggests that the quotient is indeterminant.
x/0x/0 Is Undefined or Indeterminant
Division by 0 is undefined or indeterminant. Do not divide by 0.
Rational numbers have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are:
Irrational Numbers
The irrational numbers (Ir)(Ir): Irrational numbers are numbers that cannot be written as the quotient of two integers. They are numbers whose decimal representations are nonterminating and nonrepeating. Some examples are
4.01001000100001…π=3.1415927…4.01001000100001…π=3.1415927…
Notice that the collections of rational numbers and irrational numbers have no numbers in common.
When graphed on the number line, the rational and irrational numbers account for every point on the number line. Thus each point on the number line has a coordinate that is either a rational or an irrational number.
Exercise 11
Solution
Sample Set C
Example 4
What integers can replace xx so that the following statement is true?
−4≤x<2−4≤x<2
This statement indicates that the number represented by xx is between −4−4 and 2. Specifically, −4−4 is less than or equal to xx, and at the same time, xx is strictly less than 2. This statement is an example of a compound inequality.
The integers are −4,−3,−2,−1,0,1−4,−3,−2,−1,0,1.
Example 5
Draw a number line that extends from −3−3 to 7. Place points at all whole numbers between and including −2−2 and 6.
Example 6
Draw a number line that extends from −4−4 to 6 and place points at all real numbers greater than or equal to 3 but strictly less than 5.
It is customary to use a closed circle to indicate that a point is included in the graph and an open circle to indicate that a point is not included.
Practice Set C
Exercise 12
What whole numbers can replace xx so that the following statement is true? −3≤x<3−3≤x<3
Exercise 13
Solution
Exercises
For the following problems, next to each real number, note all collections to which it belongs by writing NN for natural numbers, WW for whole numbers, ZZ for integers, QQ for rational numbers, IrIr for irrational numbers, and RR for real numbers. Some numbers may require more than one letter.
Exercise 21
Exercise 22
Solution
Exercise 23
Exercise 24
Solution
Exercise 25
An integer is an even integer if it can be divided by 2 without a remainder; otherwise the number is odd. Draw a number line that extends from −5−5 to 5 and place points at all negative even integers and at all positive odd integers.
Exercise 26
Draw a number line that extends from −5−5 to 5. Place points at all integers strictly greater than −3−3 but strictly less than 4
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0534402305David Cohen's PRECALCULUS, WITH UNIT-CIRCLE TRIGONOMETRY, Fourth Edition, focuses on teaching mathematics, using a graphical perspective throughout to provide a visual understanding of college algebra and trigonometry. The author is known for his clear writing style and the numerous quality exercises and applications he includes in his respected texts. In this new edition, graphs, visualization of data, and functions are now introduced much earlier and receive greater emphasis. Many sections now contain more examples and exercises involving applications and real-life data. While this edition takes the existence of the graphing calculator for granted, the material is arranged so that one can teach the course with as much or as little graphing utility work as he/she wishes.
Editorial Reviews
From the Publisher
"The quantity and quality of the exercises are the reasons why I think that Cohen's book is my favorite algebra and precalculus book ever…. Both the quantity and quality of the exercises of Cohen's book are first class and the problems in each section are reflective of the concepts being taught. As far as I am concerned, the quality of both his algebra text and precalculus text are way above anything on the market that I've seen."
"The mini projects are excellent and can be used for collaborative learning…. I think the writing style of an already well-written book has been improved."
"This text is one of the most challenging precalculus books I have seen, but I like that because it gives students the opportunity to be very well prepared for calculus. The presentations are rigorous, precise, and detailed, but written in a way that students can follow fairly well. The exercises range from simple to complex and include not only applications to other disciplines but to other areas of mathematics as well." "One of the main reasons I continue to use this author's texts is that I receive more compliments from my students on his books than I have on any other texts."
"Overall, the quantity and quality of the exercises in Cohen's texts are outstanding. The different levels of exercises—A, B, and C levels—make an excellent transition from skill development to concept development."
"One of the strengths of the series has been David's ability to make simple, clear, and reasoned arguments about the mathematical assertions…. David did a very subtle and powerful thing when he included difficult problems broken down into small, manageable steps." "The new authors have done a nice job in reorganizing the material in Chapter 6. Instructors familiar with earlier editions of Cohen's books will find the transition seamless."
Related Subjects
Meet the Author
David Cohen, a senior lecturer at UCLA, was the original author of the successful, well-respected precalculus series—COLLEGE ALGEBRA, ALGEBRA AND TRIGONOMETRY, PRECALCULUS: A PROBLEMS-ORIENTED APPROACH, and PRECALCULUS: WITH UNIT CIRCLE TRIGONOMETRYWritten for profit not for learning
I was required to use this text for my pre-calc class and was completely disappointed. First, it costs $150. What has changed in calculus in 200 years that requires so many editions? A well written text should work forever but that doesn't help the publisher's profits; you can figure out the rest. Second it only gives the answers to odd number problems. At this level math the most important thing is practice, you only have access to half the practice. With only half the answer shouldn't the book only cost $75, at least until they provide the other answers to the practice questions. Third, the text is written from the standpoint of someone who understands the subject explaining it to someone else who understands the subject. It needs to be written for someone who DOESN'T understand the subject. Math departments would do well to stay away from this book because choosing such a bad text reflects the ability of that department to make decisions. If a board member chooses this text, their bank account should be audited to find kickbacks from the publisher.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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Trade in Cambridge IGCSE Mathematics (International GCSE) for an Amazon.co.uk gift card of up to £0.40, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book DescriptionThis is an excellent resource for anyone taking the Cambridge IGCSE. The explanations are clear and concise with enough examples to illustrate the various methods that can be used to solve a problem. The CD-Rom that comes with the book is also a useful tool with which to practise everything learnt. The only proviso I would make is that the student would need to find some more practice papers to complete before taking the exam.
This Maths book is excellent for visual learners. It has lots of diagrams and careful explanations on how to do the sums. It simplifies most of the explanations which is really good for struggling students. It covers the full syllabus for Core Maths and introduces some Extended topics. Every question in the book is answered; even the graph and Diagram questions have answers provided so if you have a curve graph you won't need to wonder if it's correct. It's the only Maths book I know that has some colour diagrams which seems to help my daughter with concentration. If you are doing Extended Maths you will need another book for extra practice as the questions in this book are limited to about 5 a chapter. This book and the David Rayner books (either Extended or Core) perfectly complement one another. Collins explains the execution, while Rayner give you tons of practice. I did not like the way the CD was structured as instead of giving PDF worksheets of extra revision practice, it's an interactive interface which launches your Internet browser and asks you very simple questions; simpler than the book and hardly useful. They are ok for simple quick revision but not challenging enough to sharpen your skills for a good grade. Sample questions from the CD are:
Write 0.5 as a fraction in the form of a/b. Another: solve the equation x+5=8.
Hardly challenging. But the book itself is still worth buying as a main text for Core and supplementary for Extended.
Go through this book chapter by chapter doing the short pracice excercises at the end of each section. It has clear concise examples. I found that the course books went into so much detail that could have been explained in more simple ways. I agree with angie. Try and get hold of as many practice papers you can to get extra practice. Good Luck!
The book brings the mathematical rules and principles into a nice digestible form. Well done, especially for the novice in Math. Also excellent for the intermediate person reviewing their past Math knowledge.
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Created by Lang Moore and David Smith for the Connected Curriculum Project, this is a module to review concepts of inverse functions, and to use those concepts, together with functions defined by integrals, to developThis lesson from Illuminations teaches students the features of sine and cosine graphs. Students use uncooked spaghetti to demonstrate the properties of the unit circle, which they will then represent with graphs. It is...
This pdf contains a syllabus for a first course on structural fabrication as part of the Aerospace Technology Program. Topics include trigonometry, machine tools, blueprints, metal working, aerospace fasteners, and...
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Math-drills.com - math worksheets | free printable math, Over 21,000 math worksheets for multiplication, addition, fractions, decimals, geometry, measurement and many other math topics..
Mathematics authors/titles "new" - arxiv, We study several properties of commutative local rings that generalize the notion of koszul algebra. the properties are expressed in terms of the ext algebra of the.
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Easy Algebra Step-By-Step
Take it step-by-step for algebra success! The quickest route to learning a subject is through a solid grounding in the basics. So what you won't find ...Show synopsisTake it step-by-step for algebra success! The quickest route to learning a subject is through a solid grounding in the basics. So what you won't find in Easy Algebra 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 that are linked to core skills--enabling learners to grasp when and how to apply those techniques. This book features: Large step-by-step charts breaking down each step within a process and showing clear connections between topics and annotations to clarify difficulties Stay-in-step panels show how to cope with variations to the core steps Step-it-up exercises link practice to the core steps already presented Missteps and stumbles highlight common errors to avoid You can master algebra as long as you take it Step-by-Step
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Therefore, by the end of the level of pre algebra, students are able to be adept at manipulating numbers and equations and understand the general principles at work. Students make numbers sense and use factoring of numerators and denominators and properties of exponents. Factoring a numbers is ...
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Featured Webinar
Now that you've received your evaluation copy of Maple, you may be wondering what you can do with it! This webinar, presented by Dr. Robert Lopez, Maple Fellow and Emeritus Professor from the Rose Hulman Institute of Technology, will provide you with tips and techniques that will help you get started with Maple 18.
This webinar offers educators a quick and easy way to learn some of the fundamental concepts of Maple. Learn a few simple techniques that will allow you to use Clickable Math™ features to compose, visualize, and solve a wide variety of mathematical problems without commands. This webinar will also provide an introduction to some of the technical documentation features in Maple, including the use of interactive components such as buttons and sliders.
During this webinar Maplesoft will present a number of examples of mathematics in film. See relevant, exciting examples that you can use to engage your students. Have you ever wondered if the bus could really have jumped the gap in "Speed?" We've got the answer! Anyone with an interest in mathematics, especially high school and early college math educators, will be both entertained and informed by attending this webinar. At the end of the webinar you'll be given an opportunity to download an application containing all of the Hollywood examples that we demonstrate.
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Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one-one, into and onto functions, composition of functions.
UNIT 2: COMPLEX NUMBERS AND QUADRATIC EQUATIONS:
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots.
UNIT 3: MATRICES AND DETERMINANTS:
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three.
Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.
UNIT 4: PERMUTATIONS AND COMBINATIONS:
Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications.
UNIT 5: MATHEMATICAL INDUCTION:
Principle of Mathematical Induction and its simple applications.
UNIT 6: BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS
Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.
UNIT 7: SEQUENCES AND SERIES:
Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers. Relation between A.M. and G.M. Sum upto n terms of special series: S n, S n2, Sn3.
Evaluation of simple integrals of the type Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.
UNIT 10: DIFFERENTIAL EQUATIONS:
Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations of the type:
dy+ p (x) y = q (x)
dx
UNIT 11: CO-ORDINATE GEOMETRY:
Cartesian system of rectangular co-ordinates 10 in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines
Various forms of equations of a line, intersection of lines, angles betweentwo lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines.
Circles, conic sections
Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.
UNIT 12 THREE DIMENSIONAL GEOMETRY:
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines.
Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.
UNIT 13: VECTOR ALGEBRA:
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
UNIT 14: STATISTICS AND PROBABILITY:
Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data calculation of standard deviation, variance and mean deviation for grouped and ungrouped data.
Rate of a chemical reaction, factors affecting the rate of reactions: concentration, temperature, pressure and catalyst; elementary and complex reactions, order and molecularity of reactions, rate law, rate constant and its units, differential and integral forms of zero and first order reactions, their characteristics and half – lives, effect of temperature on rate of reactions –
Modes of occurrence of elements in nature, minerals, ores; Steps involved in the extraction of metals – concentration, reduction (chemical and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe; Thermodynamic and electrochemical principles involved in the extraction of metals.
UNIT 13: HYDROGEN
Position of hydrogen in periodic table, isotopes, preparation, properties and uses of hydrogen; Physical and chemical properties of water and heavy water; Structure, preparation, reactions and uses of hydrogen peroxide; Hydrogen as a fuel.
UNIT 14: S – BLOCK ELEMENTS (ALKALI AND ALKALINE EARTH METALS)
Group – 1 and 2 Elements
General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous properties of the first element of each group, diagonal relationships.
Preparation and properties of some important compounds – sodium carbonate and sodium hydroxide; Industrial uses of lime, limestone, Plaster of Paris and cement; Biological significance of Na, K, Mg and Ca.
UNIT 15: P – BLOCK ELEMENTS
Group – 13 to Group 18 Elements
General Introduction: Electronic configuration and general trends in physical and chemical properties of elements across the periods and down the groups; unique behaviour of the first element in each group.
Aldehyde and Ketones: Nature of carbonyl group;Nucleophilic addition to >C=O group, relative reactivities of aldehydes and ketones; Important reactions such as – Nucleophilic addition reactions (addition of HCN, NH3 and its derivatives),
The JEE Main 2015 Syllabus for Physics contains two Sections – A and B. Section – A pertains to the Theory Part having 80% weightage, while Section – B contains Practical Component (Experimental Skills) having 20% weightage.
Force and Inertia, Newton's First Law of motion; Momentum, Newton's Second Law of motion; Impulse; Newton's Third Law of motion. Law of conservation of linear momentum and its applications, Equilibrium of concurrent forces.
Static and Kinetic friction, laws of friction, rolling friction.
Dynamics of uniform circular motion: Centripetal force and its applications.
UNIT 4: WORK, ENERGY AND POWER
Work done by a constant force and a variable force; kinetic and potential energies, workenergy theorem, power.
Potential energy of a spring, conservation of mechanical energy, conservative and non-conservative forces; Elastic and inelastic collisions in one and two dimensions.
UNIT 5: ROTATIONAL MOTION
Centre of mass of a two-particle system, Centre of mass of a rigid body; Basic concepts of rotational motion; moment of a force, torque, angular momentum, conservation of angular momentum and its applications; moment of inertia, radius of gyration. Values of moments of inertia for simple geometrical objects, parallel and perpendicular axes theorems and their applications. Rigid body rotation, equations of rotational motion.
UNIT 6: GRAVITATION
The universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Kepler's laws of planetary motion. Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity of a satellite. Geo-stationary satellites.
Thermal equilibrium, zeroth law of thermodynamics, concept of temperature.
Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics: reversible and irreversible processes. Carnot engine and its efficiency.
UNIT 9: KINETIC THEORY OF GASES
Equation of state of a perfect gas, work doneon compressing a gas.Kinetic theory of gases – assumptions, concept of pressure. Kinetic energy and temperature: rms speed of gas molecules; Degrees of freedom, Law of equipartition of energy,applications to specific heat capacities of gases; Mean free path, Avogadro's number.
Electric field: Electric field due to a point charge, Electric field lines, Electric dipole, Electric field due to a dipole, Torque on a dipole in a uniform electric field.
Electric flux, Gauss's law and its applications to find field due to infinitely long uniformly charged straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell. Electric potential and its calculation for a point charge, electric dipole and system of charges; Equipotential surfaces, Electrical potential energy of a system of two point charges in an electrostatic field.
Conductors and insulators, Dielectrics and electric polarization, capacitor, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, Energy stored in a capacitor.
Electric Cell and its Internal resistance, potential difference and emf of a cell, combination of cells in series and in parallel. Kirchhoff's laws and their applications. Wheatstone bridge, Metre bridge. Potentiometer – principle and its applications.
UNIT 13: MAGNETIC EFFECTS OF CURRENT AND MAGNETISM
Biot – Savart law and its application to current carrying circular loop. Ampere's law and its applications to infinitely long current carrying straight wire and solenoid. Force on a moving charge in uniform magnetic and electric fields. Cyclotron.
Force on a current-carrying conductor in a uniform magnetic field. Force between two parallel current-carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field; Moving coil galvanometer, its current sensitivity and conversion to ammeter and voltmeter.
Current loop as a magnetic dipole and its magnetic dipole moment. Bar magnet as an equivalent solenoid, magnetic field lines; Earth's magnetic field and magnetic elements.Para-, dia- and ferro- magnetic substances.
Reflection and refraction of light at plane and spherical surfaces, mirror formula, Total internal reflection and its applications, Deviation and Dispersion of light by a prism, Lens Formula, Magnification, Power of a Lens, Combination of thin lenses in contact, Microscope and Astronomical Telescope (reflecting and refracting) and their magnifyingpowers.
JEE Main 2015 Syllabus covers three subjects for paper 1 (B.Tech Engineering programmes) : Physics, Chemistry and Mathematics. Each of the syllabus subjects are discussed here under. The syllabus for JEE Main 2015 is CBSE 10+2 pattern syllabus prescribed for both class 11 and 12 students. A detailed contents of the syllabus can be found in the following linked pages:
I hereby declare that all the particulars stated in this application form are true to the best of my knowledge and belief. I have read and understood the JEE procedures for both JEE (Main)and JEE (Advanced) – 2014. I shall abide by the terms and conditions thereon.
*The option of date for Computer Based Examination for Paper – 1 should be exercised while filling up the application form.
The allotment of slots/dates will be on first come first served basis.
If a candidate does not make any selection, he/she shall be randomly assigned a slot/date as per the availability of the same.
The Computer Based Examination for Paper – 1 in Colombo, Kathmandu, Singapore, Bahrain, Dubai, Muscat and Riyadh will be held only on 19/04/2014.
In case a candidate, by furnishing the false information, appears in more than one slots/dates of the computer based examination or appears in both the modes of examination i.e. pen & paper based and computer based examination, his candidature will be cancelled and his result will not be declared.
JEE MAIN Test is the national level Under Graduate Engineering Entrance Test conducted for the first time in India. JEE MAIN 2013 is also the entry level exam for IIT JEE (JEE Advanced 2013). Students need to score within top 1,50,000 candidates to appear the JEE Advanced Exam 2013. JEE MAIN 2013 has replaced the earlier famous AIEEE Test. The test shall be conducted both online mode and pen and paper based. Admission into NITs IIITs and other Centrally Funded Institutions shall be based on JEE MAIN Score as well as Board marks obtained by the candidates. Hence, it is highly important for a student to achieve good score for building his / her engineering career. Entranceindia has released 15+1 Model Papers and Practice DVDs for JEE MAIN 2013 candidates. Students can avail these Online Test Series by logging into the system after payment of desired course fee ranging from Rs.450/- to 950/-. Each Model Test Paper contains 90 questions from subjects Physics, Chemistry and Mathematics. Students can attempt the mock paper test and review their answers with detailed solution till 2nd June 2013 the JEE Advanced Exam.
Your preparation could be more meaningful with lots of practice questions on the JEE MAIN 2013 Syllabus. You can choose to practice similar pattern questions as expected in JEE MAIN 2013 Examination. Entranceindia brought detailed Practice Papers for JEE Main 2013 candidates. Our Practice Papers can be practised online and cover JEE Main Pattern Questions from the whole syllabus prescribed for the examination. Click the below links to know more about our Online Practice Tests.
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umerical Solution of Ordinary Differential Equations
A concise introduction to numerical methodsand the mathematical framework neededto understand their performance "Numerical Solution of Ordinary ...Show synopsisA concise introduction to numerical methodsand the mathematical framework neededto understand their performance "Numerical Solution of Ordinary Differential Equations" presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in order to help readers comprehend the applications of ordinary differential equations. In addition, the authors' collective academic experience ensures a coherent and accessible discussion of key topics, including: Euler's method Taylor and Runge-Kutta methods General error analysis for multi-step methods Stiff differential equations Differential algebraic equations Two-point boundary value problems Volterra integral equations Each chapter features problem sets that enable readers to test and build their knowledge of the presented methods, and a related Web site features MATLAB(R) programs that facilitate the exploration of numerical methods in greater depth. Detailed references outline additional literature on both analytical and numerical aspects of ordinary differential equations for further exploration of individual topics. "Numerical Solution of Ordinary Differential Equations" is an excellent textbook for courses on the numerical solution of differential equations at the upper-undergraduate and beginning graduate levels. It also serves as a valuable reference for researchers in the fields of mathematics and engineering.Hide synopsis
Description:New. 047004294X ***BRAND-NEW*** FAST UPS shipping, so you'll...New. 047004294
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Intermediate Algebra - 4th edition
Summary: Algebra can be like a foreign language. But one text delivers an interpretation you can fully understand. Building a conceptual foundation in the ''language of algebra,'' iNTERMEDIATE ALGEBRA, 4e provides an integrated learning process that helps you expand your reasoning abilities as it teaches you how to read, write, and think mathematically. Packed with real-life applications of math, it blends instructional approaches that include vocabulary, practice, and well-defined pedagogy w...show moreith an emphasis on reasoning, modeling, communication, and technology skills. The authors' five-step problem-solving approach makes learning easy. More student-friendly than ever, the text offers a rich collection of student learning tools, including Enhanced WebAssign online learning system. With INTERMEDIATE ALGEBRA, 4e, algebra makes sense!Book Daddy Fort Wayne, IN
Hardcover Fair 0495389730 Item Is In Used Condition. USPS or FedEx Tracking # included with all orders. 100% Satisfaction Guarantee Coverage.
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Hardcover Very Good 0495389730 In Very Good Used Condition. USPS or FedEx Tracking # included with all orders. 100% Satisfaction Guarantee Coverage01 +$3.99 s/h
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A Book Company Lexington, KY
May contain some highlighting. Supplemental materials may not be included. We select best copy available. - 4th Edition - Hardcover - ISBN 9780495389736
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Math Smart Getting a Grip Basic Math 2ND Edition
Synopses & Reviews
Publisher Comments:
Whether an individual is checking it out at the supermarket or cashing it in at the stock market, one needs a command of basic math to survive in today's number crunching world.
But most people have problems with math. A decimal here and an exponent there and they've gone from a balanced checkbook to a multi-trillion-dollar national debt. That's why The Princeton Review created Math Smart.
Math Smart's approach is easy to follow. It will show readers how to perform basic math operations like addition, subtraction, multiplication, and division. Once they've got that down, The Princeton Review will teach them how to handle the scary stuff like exponents, square roots, geometry, and algebra.
How does Math Smart work? It teaches user-friendly techniques that break down complicated problems and equations into their basic parts. Readers won't waste their time memorizing dozens of long formulas and equations
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11. Math Cheat Sheet
11. Math Cheat Sheet
Math forms the basis of computer science in general and computer graphics in particular. The term linear algebra encompasses the majority of mathematic operations used in modern computer graphics. Other common terms for the same operations include vector algebra and matrix math. Study of vectors and vector transformations began in the 1600s and matured into its current form in the 1790s. Charles Babbage designed his mechanical general-purpose computer called the "Analytical Engine" in 1837. With the Analytical Engine in mind, Augusta Ada King, Countess of Lovelace, wrote the first computer program in 1842. As a result, when transistors were invented in 1948 to replace mechanical switches and vacuum tubes, mathematicians already knew what to do with them, and the software industry was born.
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A Survey of Mathematics with Applications64.48
FREE
Used Good
(464.98
FREE
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(76541
FREE
About the Book
In a Liberal Arts Math course, a common question students ask is, "Why do I have to know this?" A Survey of Mathematics with Applications continues to be a best-seller because it shows students how we use mathematics in our daily lives and whythis is important. The Ninth Edition further emphasizes this with the addition of new "Why This Is Important" sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills. Angel
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Find a Brookline Village PrealgebraThe typical class is trying to deal with the applications to real-world problems where the results of using mathematics are emphasized over the reasons why they work, but understanding the underlying principles can help a lot in doing well in such a course. People studying finite mathematics are...
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High School Mathematics
Mobilizing Minds: Teaching Math and Science in the Age of Sputnik
High School Mathematics
Many efforts at curriculum reform concentrated on high school teaching. Old forms of apparatus became more widely available. University professors worked with teachers to try out new approaches to teaching that emphasized the theoretical beauty of mathematics rather than its practical applications.
Demonstration Slide Rule by Keuffel & Esser, 1967
The slide rule is an instrument used to assist in multiplication, division, and other mathematical operations. Widely used in American engineering schools from about 1900, it slowly diffused into high schools and lower grades. Slide rule makers also sold oversized examples for instruction. The Winchester-Thurston School, a girl's high school in Pittsburgh, purchased this slide rule for one of its classrooms.
UICSM Programmed Textbook, 1963
At the University of Illinois, discussion of the need for new ways of teaching mathematics to high school students began in 1951. By 1963, the University of Illinois Committee on School Mathematics had developed a set of four experimental programmed textbooks. These included text in the form of questions and answers, and as tests for each section.
Blackboard Compasses, about 1950
In the 1950s and 1960s, community colleges expanded rapidly. Oversized compasses, used to draw circles and arcs of circles on the blackboard, had long been available in the United States. This instrument was used at Montgomery College in Maryland from about 1950 onward.
Experimental Units of Minnesota School Mathematics Center, 1963
Select high school students also were introduced to advanced topics in mathematics in special summer programs. Mathematician Paul C. Rosenbloom of the University of Minnesota prepared this experimental course for one of these seminars.
Students studied topics such as information theory, computer logic, set theory, calculus, and vector analysis, which had not been part of the standard high school curriculum. Students also had the opportunity to write and run programs for a CDC 160 computer. At the time, computers were extremely large and expensive, and the opportunity to use one was an unusual event. Rosenbloom and one of his colleagues later wrote a textbook that was used in many colleges.
Game of WFF'N PROOF, 1969
Professors also designed games to teach mathematical skills to people of all ages. Layman E. Allen of the Yale University Law School began thinking about improving logical thinking in 1956. From 1960, with funds from the Carnegie Corporation of New York, he spearheaded a project for the "Accelerated Learning of Logic."
Allen and his colleagues hoped to develop games and other materials that would offer a ways to convey logical principles. In 1961 and 1962 they released different versions of WFF'N PROOF, a series of 21 games designed for learners from young children to adults. Allen's brother Robert tried out the games with students in California and Florida. His success led Allen and others to revise WFF'N PROOF further.
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: Its Power and Utility
Explores the power and historic impact of mathematics and helps students harness that power by developing an effective approach to problem solving. ...Show synopsisExplores the power and historic impact of mathematics and helps students harness that power by developing an effective approach to problem solving. This title also explores the utility and application of math concepts to a wide variety of real-life situations: money management; handling of credit cards; inflation; and, many other topics.Hide synopsis
Description:Very good. Hardcover. Instructor Edition: Same as student...Very good. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Has minor wear and/or markings. SKU: 9781111581527-3. Hardcover. Instructor Edition: Same as student edition...Fine. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Almost new condition. SKU: 9781111581527-2 in very good dust jacket. Sewn binding. Cloth over boards....Fine in very good dust jacket. Sewn binding. Cloth over boards. Textbooks Available with Cengage Youbook. Audience: General/trade. ANNOTATED INSTRUCTOR'S EDITION SAME AS STUDENT EDITION WITH ANSWERS
Reviews of Mathematics: Its Power and Utility
Excellent material for pre-high school studants, or for people who want to advance their understanding of math. Includes Alegebraic problem solving, Geometry,Statistics, Logic and Sets.
Well written, lots of illustrations
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The Algebra 2 Tutor DVD Series teaches students the core topics of Algebra 2 and bridges the gap between Algebra 1 and Trigonometry, providing students with essential skills for understanding advanced mathematics.
This lesson teaches students how to solve equations that contain polynomials that cannot be easily factored. In order to do this, the quadratic formula must be used. Students are introduced to the quadratic formula and taught how to properly apply it to the equation at hand. Grades 8-12. 15 minutes on DVD.
Customer Reviews for Algebra 2 Tutor: Quadratic Formula DVD
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Topics: We will study divisibility properties of the integers, primes and their
distribution, congruences including the Chinese remainder theorem, Fermat's
theorem, Wilson's theorem, special number theoretic functions, the Euler-phi
function, the quadratic reciprocity theorem, perfect numbers, and the Fibonacci
sequence
Grading:Grades will be based on two
in-class tests, a comprehensive in-class final exam, homework, and team
problems.
Tests:The dates for the tests and
final exam are as follows:
Test 1
Tuesday, September 26
Test 2 Tuesday, November 14
Final Exam Thursday, December 7, 11A.M.- 1P.M.
Homework: There will be
regularly assigned homework (typically once per week) that will be collected
and graded. Discussion among students about the homework is permitted, and
encouraged. However, each individual must write up his or her own solutions,
which should demonstrate that individual's understanding of the work. In other
words, your written work must clearly be your own; NO ONE IS TO SHOW A SOLUTION
TO ANOTHER STUDENT BEFORE THE ASSIGNMENT IS DUE. A violation of this will
result in a grade of 0 for that entire assignment, even if only one problem is
copied (whether a student was the "copier" or "was copied from" will not
matter: a grade of 0 will be given for the homework grade).
Team Problems:Some problems
will be assigned as "team problems," where a group of 2-3 students will be
given a particular problem. These problems will be assigned on (most)
Thursdays, the teams will begin working on the problems during the latter part
of Thursday's class, and will present their solutions to the class at the next
class meeting on Tuesday. Grades for each assignment will be either a 0 or a 10
per problem, depending on whether the team is able to present their solution in
class (partial solutions are acceptable, as long as meaningful effort was
contributed by the team members). For each problem, all members of a given team
will receive the same grade (unless some member(s) did not participate). The
make-up of the teams will change from week to week.
Grading Procedure:The final
grade for the course will be determined as follows:
Two tests: 30%
Final exam 25%
Homework 25%
Team Problems 20%
Grading Scale: A: 89 or higher
B: at least 78, but less than 89
C: at least 67, but less than 78
D: at least 56, but less
than 67
F: less than 56
Make-up Test Policy: There
will be no make-up tests given
under any circumstances. If one test is missed, the final exam score will be
used in its place. If both are missed, the final exam score will replace one of
the tests, and the other test score will be 0.
Homework Policy: Homework will not be accepted later than the date it
is due. Your lowest homework score will be dropped. If a homework assignment is
late, then it will represent the "dropped" assignment. Don't be late with your
homework! (I will accept homework up to 5P.M. on the date it is due).
Academic Honesty: Homework solutions that are not your own work (even
one statement!) will result in a grade of 0 for that homework assignment. If
this occurs on a second homework assignment, the student will receive a grade
of F for the course. Any act of academic dishonesty on one of the tests or the
final exam will result in an F for the course.
Classroom rules: It is important that every student and the
instructor be able to learn and work in an atmosphere conducive to learning.
Interruptions, excessive noise, or other potentially distracting behavior
should not take place during the scheduled class time. The following are not
allowed during class time (12:30-1:45): cell phones ringing, beeping, etc.
(all cell phones must be off!); eating or drinking; classroom conversation not
relevant to class discussion; sleeping; excessive tardiness; leaving class
early without prior notice to the instructor; any other disruptive behavior.
Violations of these rules will be handled as follows: (a) 1st
violation: warning (b) 2nd or 3rd violation: 5 pts off
of final exam score for each violation; (c) more than 3 violations: student
will be removed from class and receive a W (or WF if after the deadline for
withdrawal without penalty).
Other:
It is strongly recommended
that you allow at least six hours each week, outside of class, for homework and
study (more would be better!).
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Algebra 1 Syllabus 2013-2014 Patricia Winkler
Algebra 1 is the first math course students in Texas take in the high school curriculum. It can be a difficult course and may, at times, require additional time for preparation and practice. The students are expected to come to class each day with the proper materials and the motivation to succeed in Algebra 1. It is my goal to help the students fill in gaps in their mathematics background and also to challenge them to stretch to meet the goals and objectives of this course as the foundational preparation on their journey to taking AP Calculus.
Classroom Rules:
1. The students will come to class prepared with the necessary supplies and their completed work. Due to space restrictions in the classroom, I am asking that each student bring no more than one bag into the classroom leaving unnecessary items in their lockers, for the safety of all.
2. Students need to be in the classroom before the tardy bell rings. The School Tardy Policy is in the Student Handbook.
3. Before leaving class, students will be expected to pick up and dispose of any trash in and around their work areas.
4. Students should do their own work.
5. Using class time to work on assignments for another class is not allowed, unless otherwise granted by the teacher.
6. All assignments should be handed in on time. If a student does not have an assignment completed and ready to hand in at the time it is collected, the student will receive a zero (0) for that assignment. If they hand-in the completed assignment by the beginning of the next class meeting, the grade on the assignment will be changed to a 70.
7. The students will treat others with respect and courtesy.
8. The students will refrain from eating, drinking, gum chewing, and personal grooming in class.
9. The students will refrain from sleeping in class or putting their heads down on the desks.
10. Adhere to all rules in DeBakey High School's Student Handbook and HISD's Code of Student Conduct.
Cheating of any kind will result in a zero and disciplinary action described in the Student Handbook.
The above consequences are for a violation of classroom rules or Level I rules. All violations of other levels will result in consequences as specified in the Student Code of Conduct.
Resources:
Textbook: McDougal Littell Algebra 1 2007 Texas edition. This textbook and its practice workbook are also available in electronic form, at The ACTIVATION CODE can be obtained from the teacher. Also, see your teacher for the school code to hotmath.com.
Materials:
(1) Each student should have notebook paper along with a 1" – 1.5" binder to organize all returned papers.
(2) Graph paper (either 4 or 5 squares per inch) is a necessary DAILY supply. If you don't wish to purchase graph paper, you can create and print graph paper. One online resource is at .
(3) Students should always bring a pencil to class.
(4) Many students find colored pencils/pens and highlighters to be helpful, but they are not required.
(5) Students will be assigned a numbered TI-84 graphing calculator for use in the classroom, when appropriate. Buying a graphing calculator for home use is optional, but can help build the necessary skills for future math classes. All work that REQUIRES a graphing calculator will take place in the classroom.
(6) Donations of facial tissue/Kleenex are greatly appreciated.
Grading Procedures:
The final grade a student receives will be calculated as follows: Tests will be 50% of the grade for each cycle. Quizzes and major assignments will make up 40% each cycle grade. All major assignments will be designated as quiz grades when they are assigned. Daily grades will constitute the remaining 10% of the grade. Daily work will include both classroom work and homework. Some of the daily work assignments will be graded for completion while other assignments will be collected with all or random questions to be graded for correctness. We will follow the school's general grading policy for retakes. The Math Department's homework retake policy is stated in #6 of the classroom rules. The Math Department will also drop the lowest quiz each grading cycle. Finally, if a student receives a grade on the first test of each cycle that is lower than a 70, they can schedule with the teacher to take a retake test. The original test grade and the retake test grade will be averaged together to determine the new grade for the test with a maximum grade increase to a 70.
Extra Credit:
There are no opportunities for individual extra credit in this class, so please don't ask. However, bonus questions are offered on some quizzes and tests. The teacher reserves the right to also offer extra bonus point assignments to ALL students, when appropriate.
Progress Reports:
Detailed progress reports specific to the Algebra 1 class will be given to each student during the 4th week of each grading cycle. Parents and students can also keep up-to-date on a student's progress by logging into Parent-Student-Connect at the Houston ISD Parent website. This resource parallels the teacher's gradebook and will show grades and averages in as assignments are entered. It is as close to an up-to-the-minute progress report as a parent can get. Parents and students can also set-up features which will send text messages or emails from the system if the student's grade drops below an average that they determine. Parents and students are STRONGLY ENCOURAGE to sign-up for this resource.
Homework:
A tentative schedule of lessons, homework assignments, quizzes and tests will be provided to students at the beginning of each grading cycle. Students should use this document to plan their course studies. Students should expect to have homework assigned at every class meeting. The intent of each homework assignment is to give students an opportunity to practice the skills introduced and modeled in the classroom and to give the students an idea of the types of questions that may be seen on a quiz or a test. The homework questions will usually be discussed the next class meeting. If a student does not have their homework available in class when it is called for, no credit is given. All necessary work must be shown on homework to receive credit. Therefore a sheet with just answers is not sufficient for credit, unless otherwise stated by the teacher.
If a student is present for a class lesson, the homework assignment for that lesson is due at the next class period. If a student is absent at the next class period after a lesson is given, the homework assignment will be due the day he/she returns to class.
If the student is absent from class the day a lesson is given, the student should attend the next available morning or after school tutorial to receive help. The homework assignment for the missed lesson will be due no more than 2 class days after returning from an absence.
Students who are absent on the day an assessment is given, must be prepared to makeup the assessment on the day they return to school.
Students returning from an absence on the day an assessment is given must be prepared to take the assessment as scheduled, unless the assessment covers the material that was taught the day of the absence. If so, a one class extension will be given.
The teacher may give extended extensions on an individual basis for students with extended illnesses or emergencies. Extensions will not be given for lack of organization or planning on the part of the student.
Field Lessons
All students planning to attend a Field Lesson for another subject that requires missing this class must inform me at least 3 class days in advance. They will be responsible for making up the missing instructional material on their own and schedule make up assessments (within 2-3 days after the field lesson) in consultation with me.
Cell Phone Policy
All electronic devices (cell phones, iPads, Kindles, Nooks, etc.) should be turned off and stored in student's locker or backpack during class. Students should not be listening to music, recording class lessons, taking pictures in class or searching the internet for information unless they have been given specific permission by the teacher each time the use is requested or required. If a student does not adhere to this policy the device will be confiscated and turned in to the office to be held until the following Monday. The student will have to pay the required fine at that time to get their property.
Fire Drill
During a fire drill all students must exit the classroom and go to the designated area together and wait for me to take roll. While out of the building they are to follow all school fire drill procedures or face possible disciplinary action.
Tutoring/Extra Help:
Students will have opportunities to attend tutorials outside of the school day including before school (7:10 AM -7:40 AM), Lunch and after school (3:30 PM – 4:30 PM). Details will be announced in class.
Student Keys to Success:
Paying full attention in class, taking good notes and reviewing them daily
Daily practice of previously learned concepts and working on a regular basis to learn new concepts
Working on homework assignments seriously and being prepared to ask specific questions when the homework is reviewed in class
Reviewing all quizzes and homework problems before an exam and making sure that these problems can be worked successfully without assistance
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Online Notes / Differential Equations (Notes) / First Order DE`s / Modeling with First Order DE's
Notice
Modeling with First
Order Differential Equations
We now move into one of the main applications of
differential equations both in this class and in general. Modeling is the process of writing a
differential equation to describe a physical situation. Almost all of the differential equations that
you will use in your job (for the engineers out there in the audience) are
there because somebody, at some time, modeled a situation to come up with the
differential equation that you are using.
This section is not intended to completely teach you how to
go about modeling all physical situations.
A whole course could be devoted to the subject of modeling and still not
cover everything! This section is
designed to introduce you to the process of modeling and show you what is
involved in modeling. We will look at
three different situations in this section : Mixing Problems, Population
Problems, and Falling Bodies.
In all of these situations we will be forced to make
assumptions that do not accurately depict reality in most cases, but without
them the problems would be very difficult and beyond the scope of this
discussion (and the course in most cases to be honest).
So let's get started.
Mixing Problems
In these problems we will start with a substance that is
dissolved in a liquid. Liquid will be
entering and leaving a holding tank. The
liquid entering the tank may or may not contain more of the substance dissolved
in it. Liquid leaving the tank will of
course contain the substance dissolved in it.
If Q(t) gives the amount of
the substance dissolved in the liquid in the tank at any time t we want to develop a differential
equation that, when solved, will give us an expression for Q(t). Note as well that in
many situations we can think of air as a liquid for the purposes of these kinds
of discussions and so we don't actually need to have an actual liquid, but
could instead use air as the "liquid".
The main assumption that we'll be using here is that the
concentration of the substance in the liquid is uniform throughout the
tank. Clearly this will not be the case,
but if we allow the concentration to vary depending on the location in the tank
the problem becomes very difficult and will involve partial differential
equations, which is not the focus of this course.
The main "equation" that we'll be using to model this
situation is :
Rate of
change of
Q(t)
=
Rate at
which Q(t)
enters the
tank
Rate at
which Q(t)
exits the
tank
where,
Rate of change of Q(t)
=
Rate at which Q(t) enters
the tank = (flow rate of liquid entering) x
(concentration of substance in liquid entering)
Rate at which Q(t) exits
the tank = (flow rate of liquid exiting) x
(concentration of substance in liquid exiting)
Let's take a look at the first problem.
Example 1 A
1500 gallon tank initially contains 600 gallons of water with 5 lbs of salt
dissolved in it. Water enters the tank
at a rate of 9 gal/hr and the water entering the tank has a salt
concentration of lbs/gal.
If a well mixed solution leaves the tank at a rate of 6 gal/hr, how
much salt is in the tank when it overflows?
Solution
First off, let's address the "well mixed solution"
bit. This is the assumption that was
mentioned earlier. We are going to
assume that the instant the water enters the tank it somehow instantly
disperses evenly throughout the tank to give a uniform concentration of salt
in the tank at every point. Again,
this will clearly not be the case in reality, but it will allow us to do the
problem.
Now, to set up the IVP that we'll need to solve to get Q(t) we'll need the flow rate of the
water entering (we've got that), the concentration of the salt in the water
entering (we've got that), the flow rate of the water leaving (we've got
that) and the concentration of the salt in the water exiting (we don't have
this yet).
So, we first need to determine the concentration of the
salt in the water exiting the tank.
Since we are assuming a uniform concentration of salt in the tank the
concentration at any point in the tank and hence in the water exiting is
given by,
The amount at any time t
is easy it's just Q(t). The volume is also pretty easy. We start with 600 gallons and every hour 9
gallons enters and 6 gallons leave.
So, if we use t in hours,
every hour 3 gallons enters the tank, or at any time t there is 600 + 3t
gallons of water in the tank.
So, the IVP for this situation is,
This is a linear differential equation and it isn't too
difficult to solve (hopefully). We
will show most of the details, but leave the description of the solution
process out. If you need a refresher
on solving linear first order differential equations go back and take a look
at that section.
So, here's the general solution. Now, apply the initial condition to get the
value of the constant, c.
So, the amount of salt in the tank at any time t is.
Now, the tank will overflow at t = 300 hrs. The amount of
salt in the tank at that time is.
Here's a graph of the salt in the tank before it
overflows.
Note that the whole graph should have small oscillations
in it as you can see in the range from 200 to 250. The scale of the oscillations however was
small enough that the program used to generate the image had trouble showing
all of them.
The work was a little messy with that one, but they will
often be that way so don't get excited about it. This first example also assumed that nothing
would change throughout the life of the process. That, of course will usually not be the
case. Let's take a look at an example
where something changes in the process.
Example 2 A
1000 gallon holding tank that catches runoff from some chemical process
initially has 800 gallons of water with 2 ounces of pollution dissolved in
it. Polluted water flows into the tank
at a rate of 3 gal/hr and contains 5 ounces/gal of pollution in it. A well
mixed solution leaves the tank at 3 gal/hr as well. When the amount of pollution in the holding
tank reaches 500 ounces the inflow of polluted water is cut off and fresh
water will enter the tank at a decreased rate of 2 gallons while the outflow
is increased to 4 gal/hr. Determine
the amount of pollution in the tank at any time t.
Solution
Okay, so clearly the pollution in the tank will increase
as time passes. If the amount of
pollution ever reaches the maximum allowed there will be a change in the
situation. This will necessitate a
change in the differential equation describing the process as well. In other words, we'll need two IVP's for
this problem. One will describe the
initial situation when polluted runoff is entering the tank and one for after
the maximum allowed pollution is reached and fresh water is entering the
tank.
Here are the two IVP's for this problem.
The first one is fairly straight forward and will be valid
until the maximum amount of pollution is reached. We'll call that time tm. Also, the
volume in the tank remains constant during this time so we don't need to do
anything fancy with that this time in the second term as we did in the previous
example.
We'll need a little explanation for the second one. First notice that we don't "start over" at t = 0.
We start this one at tm,
the time at which the new process starts.
Next, fresh water is flowing into the tank and so the concentration of
pollution in the incoming water is zero.
This will drop out the first term, and that's okay so don't worry
about that.
Now, notice that the volume at any time looks a little
funny. During this time frame we are
losing two gallons of water every hour of the process so we need the "-2" in
there to account for that. However, we
can't just use t as we did in the
previous example. When this new
process starts up there needs to be 800 gallons of water in the tank and if
we just use t there we won't have
the required 800 gallons that we need in the equation. So, to make sure that we have the proper
volume we need to put in the difference in times. In this way once we are one hour into the
new process (i.e t - tm =
1) we will have 798 gallons in the tank as required.
Finally, the second process can't continue forever as
eventually the tank will empty. This
is denoted in the time restrictions as te. We can also note that te = tm
+ 400 since the tank will empty 400 hours after this new process starts up. Well, it will end provided something
doesn't come along and start changing the situation again.
Okay, now that we've got all the explanations taken care
of here's the simplified version of the IVP's that we'll be solving.
The first IVP is a fairly simple linear differential
equation so we'll leave the details of the solution to you to check. Upon solving you get.
Now, we need to find tm. This isn't too bad all we need to do is
determine when the amount of pollution reaches 500. So we need to solve.
So, the second process will pick up at 35.475 hours. For completeness sake here is the IVP with
this information inserted.
This differential equation is both linear and separable
and again isn't terribly difficult to solve so I'll leave the details to you
again to check that we should get.
So, a solution that encompasses the complete running time
of the process is
Here is a graph of the amount of pollution in the tank at
any time t.
As you can surely see, these problems can get quite
complicated if you want them to. Take
the last example. A more realistic
situation would be that once the pollution dropped below some predetermined
point the polluted runoff would, in all likelihood, be allowed to flow back in
and then the whole process would repeat itself.
So, realistically, there should be at least one more IVP in the process.
Let's move on to another type of problem now.
Population
These are somewhat easier than the mixing problems although,
in some ways, they are very similar to mixing problems. So, if P(t)
represents a population in a given region at any time t the basic equation that we'll use is identical to the one that we
used for mixing. Namely,
Rate of
change of
P(t)
=
Rate at
which P(t)
enters the
region
-
Rate at
which P(t)
exits the
region
Here the rate of change of P(t) is still the derivative.
What's different this time is the rate at which the population enters
and exits the region. For population
problems all the ways for a population to enter the region are included in the
entering rate. Birth rate and migration
into the region are examples of terms that would go into the rate at which the
population enters the region. Likewise,
all the ways for a population to leave an area will be included in the exiting
rate. Therefore things like death rate,
migration out and predation are examples of terms that would go into the rate
at which the population exits the area.
Here's an example.
Example 3 A
population of insects in a region will grow at a rate that is proportional to
their current population. In the
absence of any outside factors the population will triple in two weeks
time. On any given day there is a net
migration into the area of 15 insects and 16 are eaten by the local bird
population and 7 die of natural causes.
If there are initially 100 insects in the area will the population
survive? If not, when do they die out?
Solution
Let's start out by looking at the birth rate. We are told that the insects will be born
at a rate that is proportional to the current population. This means that the birth rate can be written
as
where r is a
positive constant that will need to be determined. Now, let's take everything into account and
get the IVP for this problem.
Note that in the first line we used parenthesis to note
which terms went into which part of the differential equation. Also note that we don't make use of the
fact that the population will triple in two weeks time in the absence of
outside factors here. In the absence
of outside factors means that the ONLY thing that we can consider is birth
rate. Nothing else can enter into the
picture and clearly we have other influences in the differential equation.
So, just how does this tripling come into play? Well, we should also note that without
knowing r we will have a difficult
time solving the IVP completely. We
will use the fact that the population triples in two week time to help us
find r. In the absence of outside factors the
differential equation would become.
Note that since we used days as the time frame in the
actual IVP I needed to convert the two weeks to 14 days. We could have just as easily converted the
original IVP to weeks as the time frame, in which case there would have been
a net change of 56
per week instead of the 8
per day that we are currently using in the original differential equation.
Okay back to the differential equation that ignores all
the outside factors. This differential
equation is separable and linear and is a simple differential equation to
solve. I'll leave the detail to you to
get the general solution.
Applying the initial condition gives c = 100. Now apply the
second condition.
We need to solve this for r. First divide both sides
by 100, then take the natural log of both sides.
We made use of the fact that here to simplify the problem. Now, that we have r we can go back and solve the original differential
equation. We'll rewrite it a little
for the solution process.
This is a fairly simple linear differential equation, but
that coefficient of P always get
people bent out of shape, so we'll go through at least some of the details
here.
Now, don't get excited about the integrating factor
here. It's just like only this time the constant is a little more
complicated than just a 2, but it is a constant! Now, solve the differential equation.
Again, do not get excited about doing the right hand
integral, it's just like integrating ! Applying the initial condition gives the
following.
Now, the exponential has a positive exponent and so will
go to plus infinity as t
increases. Its coefficient, however,
is negative and so the whole population will go negative eventually. Clearly, population can't be negative, but
in order for the population to go negative it must pass through zero. In other words, eventually all the insects
must die. So, they don't survive and
we can solve the following to determine when they die out.
So, the insects will survive for around 7.2 weeks. Here is a graph of the population during
the time in which they survive.
As with the mixing problems, we could make the population
problems more complicated by changing the circumstances at some point in
time. For instance, if at some point in
time the local bird population saw a decrease due to disease they wouldn't eat
as much after that point and a second differential equation to govern the time
after this point.
Let's now take a look at the final type of problem that
we'll be modeling in this section.
Falling Body
This will not be the first time that we've looked into
falling bodies. If you recall, we looked
at one of these when we were looking at Direction
Fields. In that section we saw that
the basic equation that we'll use is Newton's
Second Law of Motion.
The two forces that we'll be looking at here are gravity and
air resistance. The main issue with
these problems is to correctly define conventions and then remember to keep
those conventions. By this we mean
define which direction will be termed the positive direction and then make sure
that all your forces match that convention.
This is especially important for air resistance as this is usually
dependent on the velocity and so the "sign" of the velocity can and does affect
the "sign" of the air resistance force.
Let's take a look at an example.
Example 4 A
50 kg mass is shot from a cannon straight up with an initial velocity of
10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v determine the velocity of the mass
when it hits the ground.
Solution
First, notice that when we say straight up, we really mean
straight up, but in such a way that it will miss the bridge on the way back
down. Here is a sketch of the
situation.
Notice the conventions that we set up for this
problem. Since the vast majority of
the motion will be in the downward direction we decided to assume that
everything acting in the downward direction should be positive. Note that we also defined the "zero
position" as the bridge, which makes the ground have a "position" of 100.
Okay, if you think about it we actually have two
situations here. The initial phase in
which the mass is rising in the air and the second phase when the mass is on
its way down. We will need to examine
both situations and set up an IVP for each.
We will do this simultaneously.
Here are the forces that are acting on the object on the way up and on
the way down.
Notice that the air resistance force needs a negative in
both cases in order to get the correct "sign" or direction on the force. When the mass is moving upwards the
velocity (and hence v) is negative,
yet the force must be acting in a downward direction. Therefore, the "-" must be part of the force
to make sure that, overall, the force is positive and hence acting in the
downward direction. Likewise, when the
mass is moving downward the velocity (and so v) is positive. Therefore,
the air resistance must also have a "-" in order to make sure that it's
negative and hence acting in the upward direction.
So, the IVP for each of these situations are.
In the second IVP, the t0
is the time when the object is at the highest point and is ready to start on
the way down. Note that at this time
the velocity would be zero. Also note
that the initial condition of the first differential equation will have to be
negative since the initial velocity is upward.
In this case, the differential equation for both of the
situations is identical. This won't
always happen, but in those cases where it does, we can ignore the second IVP
and just let the first govern the whole process.
So, let's actually plug in for the mass and gravity (we'll
be using g = 9.8 m/s2
here). We'll go ahead and divide out
the mass while we're at it since we'll need to do that eventually anyway.
This is a simple linear differential equation to solve so
we'll leave the details to you. Upon
solving we arrive at the following equation for the velocity of the object at
any time t.
Okay, we want the velocity of the ball when it hits the
ground. Of course we need to know when it hits the ground before we can
ask this. In order to find this we
will need to find the position function.
This is easy enough to do.
We can now use the fact that I took the convention that s(0) = 0 to find that c = -1080. The position at any time is then.
To determine when the mass hits the ground we just need to
solve.
We've got two solutions here, but since we are starting
things at t = 0, the negative is
clearly the incorrect value. Therefore
the mass hits the ground at t =
5.98147. The velocity of the object
upon hitting the ground is then.
This last example gave us an example of a situation where
the two differential equations needed for the problem ended up being identical
and so we didn't need the second one after all.
Be careful however to not always expect this. We could very easily change this problem so
that it required two different differential equations. For instance we could have had a parachute on
the mass open at the top of its arc changing its air resistance. This would have completely changed the second
differential equation and forced us to use it as well. Or, we could have put a river under the
bridge so that before it actually hit the ground it would have first had to go
through some water which would have a different "air" resistance for that phase
necessitating a new differential equation for that portion.
Or, we could be really crazy and have both the parachute and
the river which would then require three IVP's to be solved before we
determined the velocity of the mass before it actually hits the solid ground.
Before leaving this section let's work a couple examples
illustrating the importance of remembering the conventions that you set up for
the positive direction in these problems.
Awhile back I gave my students a problem in which a sky
diver jumps out of a plane. Most of my
students are engineering majors and following the standard convention from most
of their engineering classes they defined the positive direction as upward,
despite the fact that all the motion in the problem was downward. There is nothing wrong with this assumption,
however, because they forgot the convention that up was positive they did not
correctly deal with the air resistance which caused them to get the incorrect
answer.
So, let's take a look at the problem and set up the IVP that
will give the sky diver's velocity at any time t.
Example 5 Set
up the IVP that will give the velocity of a 60 kg sky diver that jumps out of
a plane with no initial velocity and an air resistance of . For this example assume that the positive
direction is upward.
Solution
Here are the forces that are acting on the sky diver
Because of the conventions the force due to gravity is
negative and the force due to air resistance is positive. As set up, these forces have the correct
sign and so the IVP is
The problem arises when you go to remove the absolute
value bars. In order to do the problem
they do need to be removed. This is
where most of the students made their mistake. Because they had forgotten about the
convention and the direction of motion they just dropped the absolute value
bars to get.
So, why is this incorrect?
Well remember that the convention is that positive is upward. However in this case the object is moving
downward and so v is negative! Upon dropping the absolute value bars the
air resistance became a negative force and hence was acting in the downward
direction!
To get the correct IVP recall that because v is negative then |v| = -v. Using this, the air
resistance becomes FA = -0.8v
and despite appearances this is a positive force since the "-" cancels out
against the velocity (which is negative) to get a positive force.
The correct IVP is then
Plugging in the mass gives
For the sake of completeness the velocity of the sky
diver, at least until the parachute opens, which I didn't include in this
problem is.
This mistake was made in part because the students were in a
hurry and weren't paying attention, but also because they simply forgot about
their convention and the direction of motion!
Don't fall into this mistake.
Always pay attention to your conventions and what is happening in the
problems.
Just to show you the difference here is the problem worked
by assuming that down is positive.
Example 6 Set
up the IVP that will give the velocity of a 60 kg sky diver that jumps out of
a plane with no initial velocity and an air resistance of . For this example assume that the positive
direction is downward.
Solution
Here are the forces that are acting on the sky diver
In this case the force due to gravity is positive since
it's a downward force and air resistance is an upward force and so needs to
be negative. In this case since the
motion is downward the velocity is positive so |v| = v. The air resistance is then FA =
-0.8v. The IVP for this case is
Plugging in the mass gives
Solving this gives
This is the same solution as the previous example, except
that it's got the opposite sign. This
is to be expected since the conventions have been switched between the two
examples.
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The Advanced Placement Calculus mailing list is hosted by The College Board and archived by the Math Forum. Designed for pre-college students preparing for AP tests, the level of discussion and the explicit relation to the college sequence may also make
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AP Calculus TI 89 Program - Nils Hahnfeld
Calculator program for the TI 89 which animates calculus concepts and computes answers. Covers the entire AP Calculus (AB & BC) curriculum. Screen shots and a list of functions available at the site, as well as documentation.
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AP Computer Science - College Board Online
General information about exams and courses (A and AB), links to archives of free-response questions in C++, the AP CS mailing list, development committee access, related websites, and an on-line store for College Board books. Maintained by the College
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Aplusix: Software for Learning Algebra
Software for helping students learn algebra. In English, French, Portuguese. Includes classical exercises and word problems, with various levels of assistance. It contains 400 patterns of exercises for numerical calculation, expansion, factorization,
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APlusStudent Online Learning
Interactive math lessons with graphics, animation, and audio, based on Micromedia Flash technology. Teacher-designed, standards-based lessons. Parents can monitor the progress of their children. There is also homework help and an online gallery where
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Appealing Numbers - Ivars Peterson (MathTrek)
A short history of amicable numbers - pairs in which each number is the sum of the proper divisors of the other. The smallest such pair is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to
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AP Statistics - BB&N Upper School
A course from the Buckingham Browne & Nichols School. Advanced Placement Statistics acquaints students with the major concepts and tools for collecting, analyzing, and drawing conclusions from data, featuring work on projects involving the hands-on
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Archimedes - Chris Rorres, Drexel University
A collection of information on Archimedes' life and achievements, complete with historical quotes, illustrations, and animations. The site covers Archimedes' inventions and accomplishments: Archimedes' Claw, Burning Mirrors, the Golden Crown, the Archimedes
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Archimedes Palimpsest - The Walters Art Gallery
An introduction to the people, places, times, and historical significance of the Archimedes Palimpsest, a compendium of mathematical treatises by Archimedes of Syracuse. The manuscript itself includes the only copy of the treatise Method of Mechanical
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The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, concrete examples and practical applications show how the mastery of fundamental algebraic concepts is the key to success in today's technologically advanced world. Students will also learn the development of algebraic symbolism as well as the geometric and numeric currents. Grades 5-9. 30 minutes on DVD.
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Linear Programming Basics
HOW MAY WE HELP YOU?
Optimizing LP problems
A primer on the basics of linear programming
Linear Programming Basics
Linear programming (LP) is a powerful framework for describing and solving optimization problems. It allows you to specify a set of decision variables, and a linear objective and a set of linear constraints on these variables. To give a simple and widely used example, consider the problem of minimizing the cost of a selection of foods that meets all the recommended daily nutrient guidelines. The LP model would have a set of decision variables that capture the amount of each food to buy, a linear objective that minimizes the total cost of purchasing the chosen foods, and a linear constraint for each nutrient, requiring that the chosen foods together contain a sufficient quantity of that nutrient.
Using linear algebra notation, a linear program can be described as follows:
Objective:
minimize cTx
Constraints:
A x = b (linear constraints)
l ≤ x ≤ u (bound constraints)
When described in this form, the vector x represents the decision variables, the vector c captures the linear objective function, the matrix equation Ax = b specifies the linear constraints on x, and the vectors l and u give the lower and upper bounds on x.
The set of applications of linear programming is literally too long to list. It includes everything from production scheduling to web advertising optimization to clothing manufacturing. LP touches nearly every commercial industry in some way.
LP Algorithms
The first algorithm for solving linear programming problems was the simplex method, proposed by George Dantzig in 1947. Remarkably, this 65 year old algorithm remains one of the most efficient and most reliable methods for solving such problems today.
The primary alternative to the simplex method is the barrier or interior-point method. This approach has a long history, but its recent popularity is due to Karmarkar's 1984 polynomial-time complexity proof. Interior-point methods have benefited significantly from recent advances in computer architecture, including the introduction of multi-core processors and SIMD instructions sets, and are generally regarded as being faster than simplex for solving LP problems from scratch. However, the sheer variety of different LP models, and the many different ways in which LP is used, mean that neither algorithm dominates the other in practice. Both are important in computational linear programming.
Computational Linear Programming
Given the age of these algorithms (65 years for the simplex method, and 28 years for interior point methods), you might expect that the implementation issues associated with the methods would be well understood, and that different implementations would give similar results. Surprisingly, this is far from true. Computational benchmarks across a range of models show wide performance and robustness variations between different implementations. For example, the open-source simplex solvers CLP and GLPK are on average a factor of 2.5 and 58 times slower than the Gurobi simplex solver, respectively. What explains such wide disparities between implementations of such old and well-established methods? The differences primarily come down to three factors.
Sparse Linear Algebra
The first factor is sparse linear algebra. The constraint matrices that arise in linear programming are typically extremely sparse. Sparse matrices contain very few non-zero entries. It is not unusual to find constraint matrices containing only 3 or 4 non-zero values per columns of A. The steps of both the simplex and interior-point algorithms involve a number of computations with extremely sparse matrices and extremely sparse vectors. Sparse matrices must be factored, systems of sparse linear equations must be solved using the resulting factor matrices, the factor matrices must be modified, etc. It takes years of experience in sparse numerical linear algebra and linear programming to understand the computational issues associated with building efficient sparse matrix algorithms for LP.
Dense Matrix
Sparse Matrix
Numerical Errors
The second factor is careful handling of numerical errors. Whenever you solve systems of linear equations in finite-precision arithmetic, you will always get slight numerical errors in the results. A crucial part of building an efficient LP algorithm is to design effective strategies for managing such errors — failing to do so can mean the difference between a model solving in a fraction of a second and not solving at all.
Heuristic Strategies
The third factor is developing effective heuristic strategies for making the variety of choices that arise in the course of the solution process. To give one example, the simplex algorithm must repeatedly pick one variable from among many to enter the basis. The strategy used can have a profound effect on the runtime of the algorithm. Differences between the different strategies are often quite subtle, and in many cases they are simply based on empirical observations about which schemes are most effective in practice. Again, choosing effective strategies takes years of experience.
Benchmark Results
Public benchmarks of different commercial LP solvers demonstrate the effectiveness of the approaches that Gurobi has taken for each of these issues. For both the simplex and barrier methods, the Gurobi solvers provide both higher performance and better numerical robustness than competing solvers. This difference is of course relevant when you are solving LP models, but more importantly, it also provides a more solid foundation on which to build the many algorithms that rely on LP as a subroutine. One very important example is the branch-and-bound algorithm that is used for solving Mixed Integer Programming (MIP) models.
References
If you would like more information on these methods, we refer you to the following books:
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Add song and dance to your mathematics lessons with this exercise, in which students discover the various covert mathematical relations hidden in the lyrics to the song ?The Twelve Days of Christmas.? In addition to ...
This website applies virtual reality to calculus in order to illustrate mathematical concepts more clearly to students. While many courses utilize computers via computer algebra systems and graphing tools to...
This handy program will solve any function equation entered into its screen, and provide tips for solving a similar problem on paper. After providing the answer, the program then provides further information on the key...
Rice University provides this website, which is an introduction to key concepts in Calculus, from a high school level to advanced college math. Students can learn about major Calculus subjects and teachers can use the...
Go with Alice to a wonderland of math! This website utilizes Lewis Carroll?s bright universe and most-recognizable character in order to teach mathematical concepts. Many students may feel as though they have s...
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Colma, CA AlgebraAlgebra 2 is the time in the development of our curriculum that takes the basic skills and puts them into context. More types of functions are introduced and applied during this year. It is important at this time to connect all topics to an overarching understanding of the meaning of each in depth
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...
More About
This Book
text then serves as a resource for further investigation, explanation, and clarification.
Unlike most texts, which present exercises very similar to examples, Bassarear demonstrates how real-life problems are generally complex and often filled with ambiguity and that non-routine and multistep problems are the norm. Students learn that there may be more than one way to find an answer and even more than one answer.
A special 16-page insert includes actual pages from Houghton Mifflin's best-selling K-6 basal series, demonstrating material currently used in the classroom.
Related Subjects
Meet the Author
Tom Bassarear is a professor at Keene State College in New Hampshire. He received his BA from Claremont-McKenna College, his MA from Claremont Graduate School, and was awarded an Ed.D degree from the University of Massachusetts. Tom's complementary degrees in mathematics and educational psychology have strongly influenced his convictions about education—specifically, mathematics education. Before teaching at the college level, he taught both middle school and high school mathematics. Since arriving at Keene State College, Tom has spent many hours in elementary classrooms observing teachers and working with them in school and workshop settings, plus, he has taught 4th grade math every day for a semester at a local elementary school
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Calculus for Dummies (For Dummies)
by Mark Ryan Publisher Comments
The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have this notion that calculus is impossibly difficult unless you...A History of Pi
by Petr Beckmann Publisher Comments
The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress -- and also when it didPre-Algebra Essentials for Dummies
by Mark Zegarelli Publisher Comments
Just the critical concepts you need to score high in pre-algebra This practical, friendly guide focuses on critical concepts taught in a typical pre-algebra course, from fractions, decimals, and percents to standard formulas and simple variable equations.... (read more)
How to Lie with Statistics
by Darrell Huff Synopsis
Mr. Huff's lively, human-interest treatment of the dry-as-bones subject of statistics is a timely tonic. . . . This book needed to be written, and makes its points in an entertaining and highly readable manner.Illustrator and author pool their... (read more)
The Education of T.C. Mits: What Modern Mathematics Means to You
by Lillian R. Lieber Publisher Comments
"I have studied with pleasure [this] new book. . . . Beautiful examples. . . . Illuminating. I am convinced that [Lieber's] original enterprise will get the recognition it so richly deserves."-Albert Einstein "This is quite different from any other book.... (read more)
Mathematics-Grade 4:
by School Specialty Pub Publisher Comments
With Mathematics: A Step-By-Step Approach, Grade 4 Homework Booklet students will love building their mathematics skills while completing the fun activities in this great book Divided into four steps: addition and subtraction, multiplication, division... (read more)
The Math Chat Book
by Frank Morgan Publisher Comments
Mathematics can be fun for everyone, and this book shows it. It grew out of the author's popularisation of mathematics via live, call-in TV shows and widely published articles. The questions, comments, and even the answers here come largely from the... (read more)
Painless Math Word Problems (Barron's Painless Study)
by Marcie F Abramson Publisher Comments
(back cover) Really. This isn't going to hurt at all . . . Students discover interesting ways to see patterns in math word problems, and then make the correct computations to find solutions. In the process, they work with decimals and fractions, compare... (read more)
Rapid Math Tricks & Tips: 30 Days to Number Power
by Edward H Julius Publisher Comments
Demonstrates a slew of time-saving tips and tricks for performing common math calculations. Contains sample problems for each trick, leading the reader through step-by-step. Features two mid-terms and a final exam to test your progress plus hundreds
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08176419ical Olympiad Challenges
Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals. An important feature of the work is the comprehensive background material provided with each grouping of problems.
The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems.
Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem- solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops
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Understanding the Effects of Diet and Fitness on the Human Body through Mathematical Equations and Statistical Analysis on Calorie Intake and Calories Expended, by Ronald B. Coleman, Jr.
Guide Entry to 12.03.03:
This unit gives an interactive twist to the study of human anatomy and basic algebra for students in grades 5 and 6. Science and math are generally taught as separate subjects, yet mathematics is an integral part of scientific discovery. In this unit, students will study the basic function and developmental needs of three systems in the human body: the skeletal system, the muscular system, and the cardiovascular system. Students will work individually to understand their own personal nutritional needs, and they will work collectively to analyze hypothetical situations and compare the diets of several professional and Olympic athletes. After students have a clear, practical understanding of the nutritional values of proteins, carbohydrates, fats, and sugars they will track their own diet on a daily basis. Students will read and watch short films on the importance of combining a healthy diet with consistent exercise. Subsequently, individual students will showcase their knowledge through a self-exploration project that tracks their daily caloric intake and physical activity. Students will be able to apply their newly acquired skills towards improving their own physical and mental health. Once they are familiar with the mathematics involved in calculating net caloric intake, they will use Microsoft Excel and PowerPoint to create a final presentation. The hope is that students come away from the unit understanding the importance of proper diet and fitness in preventing disease and poor health. Finally, students will reflect on their own diet and level of physical fitness and make changes to become more confident, healthy individuals.
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Welcome to MathDL Mathematical Communication
MathDL Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.
This site addresses diverse aspects of mathematical communication, including
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MATH20201 - Algebraic Structures 1
Requisites
Aims
The course unit aims to introduce basic ideas group theory with a good range of examples so that the student has some familiarity with the fundamental concepts of abstract algebra and a good grounding for further study.
Brief Description
This course unit provides an introduction to groups, one of the most important algebraic structures. It gives the main definitions, some basic results and a wide range of examples. This builds on the study of topics such as properties of the integers, modular arithmetic, and permutations included in MATH10101/MATH10111. Groups are a fundamental concept in mathematics, particularly in the study of symmetry and of number theory.
Learning Outcomes
On completion of this unit successful students will be able to:
Appreciate and use the basic definitions and properties of groups;
Command a good understanding of the basic properties for a good range of examples;
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The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn about the history of problem solving and the derivation of the algebraic equation by functional exploration and by symbolic manipulation. Grades 5-9. 30 minutes on DVD. Finally learn the language you've always wanted to learn with the Living Language Method!
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Intermediate Algebra
"Intermediate Algebra" by Baratto/Kohlmetz/Bergman is part of the latest offerings in the successful Streeter-Hutchison Series in Mathematics. By ...Show synopsis"Intermediate Algebra" by Baratto/Kohlmetz/Bergman is part of the latest offerings in the successful Streeter-Hutchison Series in Mathematics. By popular demand, we are now offering an Intermediate Algebra book in the series again. This book combines the best of earlier versions of Intermediate Algebra, along with new material requested by a cross-section of Intermediate Algebra instructors across the country. This first edition maintains the hallmark approach of encouraging the learning of mathematics by focusing its coverage on mastering math through practice. This worktext seeks to provide carefully detailed explanations and accessible pedagogy to introduce are well-organized, and clearly labeled. Vocational and professional-technical exercises have been included throughout. Repeated exposure to this consistent structure should help advance the student's skills in relating to mathematics. The book is designed for a one-semester intermediate algebra course and is appropriate for lecture, learning center, laboratory, or self-paced courses. It is accompanied by numerous useful supplements, including McGraw-Hill's online homework management system, MathZone
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An Interactive Experience: Teaching Mathematics with Mathematica
Eric Schulz, Walla Walla Community College
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you are using a browser with JavaScript or Flash disabled, please enable them now. Otherwise, please install the latest version of the
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Generates professional-looking documents that can include text, calculations, and interactive visualizations
Provides a development environment to build flexible, structured courseware
"There are other tools that on the surface appear equivalent to Mathematica, but I have yet to find anything that compares to Mathematica's breadth, depth, elegance, and consistency."
Overview
Teaching and Mathematica go hand in hand for Eric Schulz, a mathematics instructor at Walla Walla Community College. "Mathematica is more than it appears to be to the new user. It's not just a calculator. It's not just input/output. It's a tool where you can express yourself," says Schulz. "I use Mathematica to do all of my writing, in addition to lecture notes and handouts."
Schulz also creates dynamic interfaces in Mathematica to visually enhance every lesson he teaches. He says, "I regularly think about how I can explain concepts visually because I know in the classroom I have students for which that would make sense. And then I can follow up and teach in the analytical mode."
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The Panel will advise the President and the Secretary of Education on the best use of scientifically based research to advance the teaching and learning of mathematics, with a specific focus on preparation for and success in algebra.
10 11 Basis of the Panels work
Review of 16,000 research studies and related documents.
Public testimony gathered from 110 individuals.
Review of written commentary from 160 organizations and individuals
12 public meetings held around the country
Analysis of survey results from 743 Algebra I teachers
11 12 Two Major Themes
First Things First
- Positive results can be achieved in a reasonable time at accessible cost by addressing clearly important things now.
- A consistent, wise, community-wide effort will be required.
Learning as We Go Along - In some areas, adequate research does not exist. - The community will learn more later on the basis of carefully evaluated practice and research. - We should follow a disciplined model of continuous improvement. 12 13 Curricular Content
Streamline the Mathematics Curriculum in Grades PreK-8
Follow a Coherent Progression, with Emphasis on Mastery of Key Topics
Focus on the Critical Foundations for Algebra
- Proficiency with Whole Numbers
- Proficiency with Fractions
Particular Aspects of Geometry and Measurement
Avoid Any Approach that Continually Revisits Topics without Closure
13 14 Curricular Content
An Authentic Algebra Course
All school districts
Should ensure that all prepared students have access to an authentic algebra course, and
Should prepare more students than at present to enroll in such a course by Grade 8.
14 15 Curricular Content
What Mathematics Do Teachers Need to Know
For early childhood teachers
Topics on whole numbers, fractions, and the appropriate geometry and measurement topics in the Critical Foundations of Algebra
For elementary teachers
All topics in the Critical Foundations of Algebra and those topics typically covered in an introductory Algebra course
For middle school teachers
- The Critical Foundations of Algebra
- All of the Major Topics of School Algebra
15 16 Learning Processes
Scientific Knowledge on Learning and Cognition Needs to be Applied to the Classroom to Improve Student Achievement
Most children develop considerable knowledge of mathematics before they begin kindergarten.
Children from families with low incomes, low levels of parental education, and single parents often have less mathematical knowledge when they begin school than do children from more advantaged backgrounds. This tends to hinder their learning for years to come.
There are promising interventions to improve the mathematical knowledge of these young children before they enter kindergarten.
16 17 Learning Processes
To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, factual knowledge and problem solving skills.
Limitations in the ability to keep many things in mind (working-memory) can hinder mathematics performance.
Practice can offset this through automatic recall, which results in less information to keep in mind and frees attention for new aspects of material at hand.
Learning is most effective when practice is combined with instruction on related concepts.
Conceptual understanding promotes transfer of learning to new problems and better long-term retention.
17 18 Learning Processes
Childrens goals and beliefs about learning are related to their mathematics performance.
Childrens beliefs about the relative importance of effort and ability can be changed.
Experiential studies have demonstrated that changing childrens beliefs from a focus on ability to a focus on effort increases their engagement in mathematics learning, which in turn improves mathematics outcomes.
18 19 Instructional Practices Instructional practice should be informed by high quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers.
All-encompassing recommendations that instruction should be student-centered or teacher-directed are not supported by research.
19 20 Instructional Practices
Research on students who are low achievers, have difficulties in mathematics, or have learning disabilities related to mathematics tells us that the effective practice includes
Explicit methods of instruction available on a regular basis
Clear problem solving models
Carefully orchestrated examples/ sequences of examples.
Concrete objects to understand abstract representations and notation.
Participatory thinking aloud by students and teachers.
20 21 For More Information
Please visit us online at
http//
21 22 Mathematical Proficiency Defined
National Research Council (2002) defines proficiency as
Understanding mathematics
Computing Fluently
Applying concepts to solve problems
Reasoning logically
Engaging and communicating with mathematics
23
Grous and Ceulla (2000) reported the following can increase student learning and have a positive effect on student achievement
Increasing the extent of the students opportunity to learn (OTL) mathematics content.
Focusing instruction on the meaningful development of important mathematical ideas.
Providing learning opportunities for both concepts and skills by solving problems.
Giving students both an opportunity to discover and invent new knowledge and an opportunity to practice what they have learned.
Using small groups of students to work on activities, problems, and assignments (e.g., small groups, Davidson, 1985 cooperative learning, Slavin, 1990 peer assisted learning and tutoring, Baker, et al., 2002).
Whole-class discussion following individual and group work.
Teaching math with a focus on number sense that encourages students to become problem solvers in a wide variety of situations and to view math as important for thinking.
Use of concrete materials on a long-term basis to increase achievement and improve attitudes toward math.
25 Lets turn to Alabama and Georgia 26 Alabama SBR Math SPDG-Supported Activities 27 GOAL 1 Through the implementation of SBR instructional strategies within the framework, there will be a 20 percent reduction in the achievement gap between students with and without disabilities in the area of math and age appropriate progress in pre-literacy/reading and math. 28 Alabama State Department
MATH INITIATIVE
2008-2009
29 Overview
12 school districts participated in 2007-2008. An additional 4 school districts participated in 2008-2009 (16 total).
On average, Eighth Grade students increased their Computational Fluency scores from 28.8 to 35.4.
The percent of students needing intensive focus on computational fluency decreased from 20 to 14.
61 Eighth Grade ModulesSpecial Education Students 62 Transitional Math Four school improvement schools were selected during Year 2 for implementation of Transitional Math One high school in Butler County - Greenville One high school in Elmore County - Stanhope Two high schools in Montgomery County Jefferson Davis and Robert E. Lee The four participating schools received eight days of technical assistance a month from two consultants from SOPRIS West. 63
Transitional Mathematics is designed to help students understand operations on whole numbers conceptually and addresses the needs of struggling students who have scored at or below the 40th percentile on national math tests.
Transitional Mathematics is based on three broad design principals
Ensuring that students have relevant background knowledge.
Using a balanced approach in computational practice.
Addressing the need for careful time management.
64
I. Process Evaluation
The Transitional Math program uses curriculum based student progress monitoring, which services as a fidelity tool. In August 2009, the TransMath Online Assessment System will be launched as
Individualized student placement based on students mastery of foundational math skills.
71 of all students progressed from the Frustration to Instructional or Mastery Level
66 of SWD progressed from the Frustration to Instructional or Mastery Level
CONCEPTS/ESTIMATION
Of the targeted group of students
28 were SWD
56 of all students progressed from the Frustration to Instructional or Mastery Level
45 of SWD progressed from the Frustration to Instructional or Mastery Level
101 Formative Data Examples
Coffee County Middle School
Saturday school with math focus
Math vocabulary and fluency
AIMSWeb for progress monitoring 6th and 8th gr.
Numeracy coaches
Strategies from SPDG training
Results for 24 sections of 6th grade math
79 of the sections had gt50 of students with matched scores from January to March improved
102 (No Transcript) 103 Coffee County Examining Teacher Practices
Pilot Survey of 6th Grade Teachers
Use of 12 targeted strategies from Riccominis training on differentiating in math
Six teachers participated in the survey
Twelve strategies/methods from the training were identified on the survey
104 Instruction Methods/Strategies on Survey
Grouping
Scaffolded Instruction
General Learning Strategies (Ex. RIDE)
Math Vocabulary
Spaced Instructional Review (SIR)
Interleave Worked Example
Writing about Math
Graphic Organizers for Math
Mnemonic Strategy
Fluency
Explicit Methods of Instruction
Memory Strategies
Chunking Keyword
105 Survey Results 106 2009 Statewide CRCT Results
6th Grade All Students
75 met/exceeded the standard
6 percentage point increase from 2008
15 percentage point increase since 2006
Exceeded state target
7th Grade All Students
84 met/exceeded the standard
4 percentage point increase from 2008
14 percentage point increase since 2006
Exceeded state target
8th Grade All Students
70 met/exceeded the standard
8 percentage point increase from 2009
Exceeded target
107 Students with Disabilities
CRCT Math Scores 08 to 09
More than a five percentage point increase in math scores for grades 6, 7, and 8 for SWD
108 Students with Disabilities
Georgia High School Graduation Test
Grade 11, first-time test takers
08 to 09 for SWD
63 met/exceeded standards
4 percentage point increase from 2008
109 Lessons Learned/Next Steps
Review of requirements for data collection to better ensure uniformity
Importance of continuing connection with general education statewide math initiatives
Selection of new cohort of schools for Year 3
Continued follow-up for cohort 1
other
110 Open Discussion
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Math 104: Calculus
About this Course
Math 104: Calculus is designed to prepare you to earn real college credit by passing the Calculus CLEP and Calculus Excelsior exams. This course covers topics that are included on the exams, such as polynomials, factoring, higher-order derivative and intermediate value theorem. Use it to help you learn what you need to know about calculus topics to help you succeed on the exams.
The calculus instructors are experienced and knowledgeable educators who have put together comprehensive video lessons in categories ranging from breaking a complex concept down into its basic components to calculating velocity. Each category is broken down into smaller chapters that will cover topics more in-depth. These video lessons make learning fun and interesting. You get the aid of self-graded quizzes and practice tests to allow you to gauge how much you have learned.
Category
Objectives
Applications of Derivatives
Learn how to estimate function values using linearization and how to use Newton's method to find roots of equations. Also, study linearization of functions, optimization and differentiation, optimizing complex systems and optimizing simple systems.
Area Under the Curve and Integrals
Take a look at average value theorem, definite integrals, indefinite integrals as anti-derivatives, linear properties of definite integrals, the fundamental theorem of calculus, sum notation and the trapezoid rule. Learn how to find the arc length of a function, find the limits of Riemann sums, to identify and draw left, right and middle sums, use Riemann sums for functions and graphs and use Riemann sums to calculate integrals.
Calculating Derivatives and Derivative Rules
Discover how to apply the rules of differentiation to calculate derivatives, calculate derivatives of exponential equations and find derivatives of implicit functions. Additionally, take a look at derivatives of polynomial equations, derivatives of trigonometric functions, higher-order derivatives, derivatives of inverse trigonometric functions and linear properties of a derivative.
Continuity
Study continuity in a function, discontinuities in functions and graphs, intermediate value theorem and regions of continuity in a function.
Differential Equations
Find out about differential notation in physics, separation of variables to solve system differential equations and calculating rate and exponential growth.
Geometry and Trigonometry in Calculus
Learn to find distance with the Pythagorean theorem, calculate the volumes of basic shapes and solve visualizing geometry problems. Also study sine and cosine.
Graphing and Functions
Discover concepts that include compounding functions and graphing functions of functions, figuring the equation of a line using point-slope formula, graphing exponentials and logarithms and graphing basic functions. Additionally, study horizontal and vertical asymptotes, implicit functions, exponents, slopes and tangents.
Graphing Derivatives and L'Hopital's Rule
Learn to apply L'Hopital's Rule and about concavity and inflection points on graphs, function properties from derivatives, identifying functions from derivative graphs, graphing the derivative from any function, determining maximum and minimum values of a graph and non-differentiable graphs of derivatives.
Integration and Integration Techniques
Examine anti-derivatives, integrals of simple shapes, integrals of exponential functions, integrals of trigonometric functions and improper integrals. Also take a look at how to solve integrals using substitution, how to use trigonometric substitution to solve integrals and how to factorize fractions with quadratic denominators.
Integration Applications
Learn how to calculate volumes using single intervals, find area between functions with integration, find simple areas with root finding and integration and find volumes of revolution with integration.
Limits
Study concepts including asymptotes, infinity, limits, continuity and the squeeze theorem, Also learn to determine the limits of functions and use a graph to define limits.
Rate of Change
Examine derivatives, Rolle's theorem, velocity and the rate of change. Additionally, look at the definition of mean value theorem and 'differentiable.'
Using Scientific Calculators for Calculus
Discover how to use a scientific calculator, solve equations on a scientific calculator and understand radians and degrees on a scientific calculator. Also study trigonometry functions and exponentials on a calculator
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
| 677.169 | 1 |
vision, the geometric laws that relate different views of a scene. Geometry, one of the oldest branches of ... multipleviews of a scene from the perspective of various types of geometries. A key feature is that it ... role incomputer communications. Producers and users of images, in particular three-dimensional images, ...
numerous computervision algorithms included in the OpenCV library. You will learn how to read, write, ... a variety of computervision algorithms and be exposed to important concepts in image analysis that will ... mathematical morphology and image filtering. The detection and use of interest points incomputervision is ...
filtering. The detection and use of interest points incomputervision is presented with applications for ... Exploit the image geometryin order to match different views of a pictured scene Calibrate the camera from ... programming. It can be used as a companion book in university-level computervision courses. It constitutes an ...
image pairs and of multi-view image recordings. Scientists thus began to look at basic computervision ... last decade of the twentieth century, computervision made considerable progress towards the ... consolidation of its fundaments, in particular regarding the treatment of geometry for the evaluation of stereo ...
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by Step Approach, 6thedition. This softcover edition includes all the features of the longer book, ... Browse A STEP BY STEP APPROACH is for general beginning statistics courses with a basic algebra prereq-
Textbook: ElementaryStatistics: A Brief Version, 6thedition, by Allan G. Bluman. Supplies: Any scientific calculator (a graphing calculator is not necessary.) Calculators on cell phones, tablets, or other electronic devices will NOT be allowed during tests or in-class assignments.
... particularly for the Academic Support Classes that support the 6th, ... ElementaryStatistics: Picturing the World. Larson Statistical Reasoning for ... Smith ElementaryStatistics: A Step by Step Approach. Bluman Discrete Mathematics and its Applications.
Textbook: ElementaryStatistics a Step by Step Approach, 6thEdition by Allan Bluman, McGraw/Hill, 2006. Available at Alsheqary bookstore ... Introductory statistics is a not an easy course and much of the material needs to be
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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.
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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.
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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.
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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
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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
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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
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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.
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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.
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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
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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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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.
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Physical Review Special Topics - Physics Education Research ® is a trademark of the American Physical Society.
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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.
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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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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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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
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* 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
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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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Which brings up the question of how do you get students to actually DO
mathematics? If you have to hide the answer from the student one has to
wonder if they are doing mathematics rather than repeating what they have
memorized or applying a technique they learned to a problem guaranteed to be
adapted to the technique.
Can Mathematica be used to broaden the domain of students who can start
doing some real mathematics? It seems intuitive that Mathematica can play an
important role here just because of the ease of doing the drudge work and
exploring various cases. (Could one be a good mathematician without being an
accurate paper and pencil calculator?) Just using Mathematica to automate
old techniques is not going to achieve this aim. So some suggestions:
How about giving students packages that have hierarchical depth so they can
calculate everything, at all levels, while doing derivations, proofs or
calculations? How about providing convenience routines that are adapted to
the area of interest? WRI is not great at this, and they don't have the time
for it, but they have provided the underlying capabilities for it. There is
plenty of room for people to add capability here. A requirement is that
students must be reasonably capable with Mathematica when they come to it.
How about getting away from standardized tests and having students write
mathematical essays on topics small or large? Mathematica is great at this
kind of thing. There may be no fixed answer and students may go in different
directions and even run into insurmountable problems.
How about letting students rediscover existing mathematics? Of course, they
may rediscover it on the internet so there is a different option. How about
letting students clarify existing proofs or topics? Perhaps they could
expand on difficult steps in a proof. Perhaps they could correlate a visual
proof with a formal proof. Given Mathematica's graphics, active calculation,
dynamics and control structures they might find whole new ways to present
and clarify established mathematical ideas. They can't do this without
understanding the mathematics. It requires innovation and it is definitely
value added.
How about letting students pick their own subject related topic? How about
letting groups of students work on a notebook together?
Maybe all this does not fit in with modern automated mass produced
education. I'm not an expert on it, having never taught anyone anything in
my whole life. I just comment from the perspective of a poor student.
David Park
djmpark at comcast.net
From: Murray Eisenberg [mailto:murray at math.umass.edu]
Back when I was doing such things with student projects in Mathematica,
I sure wish I had had use of David Park's HiddenNotebookData function
from his Presentations application: it would have simplified doing a lot
of things.
(But not all, probably: if you want to test a students definition of a
function on-the-fly against randomized input data, you need to hide a
"correct" definition of that function as well as generation of the test
data. At that point it may be just simpler to use a separate encoded
package of the sort I described in another post on this topic. Doing
this with a whole suite of student functions, each tested against a
series of test data, would likely create so many strings from
HiddenNotebokData that one would want to keep all that separate from the
student's own notebook where she was developing and testing the functions.)
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Math 312--Our Book: Table of the integers 1 through 400 with
their prime factorizations, the sum of their divisors, expressed as sums of
squares, and as sums of primes. Chapter 1. Elementary
facts about positive integers. Chapter 2. The
beginnings of divisibility (Other than "indivisibility," can you
think of a word with more i's than this?) theory and some random interesting
facts about triangular numbers, Pythagorean triangles, and perfect numbers. Chapter 3. Mersenne and Fermat primes,
nailing down the square triangles, counting primes, and considering polynomials
that generate them. Chapter 3.5 A characterization of
all abundant numbers with fewer than four factors. Chapter 4. Pythagorean triples,
introduction to congruence, solving linear congruences and systems of them. Chapter 5. The
distribution of primes, Fermat's Little Theorem, and sundry other good stuff. Chapter 6. The proof
of the Chinese Remainder Theorem, Wilson's Theorem, multiplicative functions. Chapter 7. Euler's
phi function and Bertrand's Postulate. Chapter 8. Some notes on the exam and primitive roots. Chapter 9. Some
elementary group theory, stuff about primitive roots, quadratic residues, and
final exam topics. Chapter 10. The Law
of Quadratic Reciprocity and sums of two squares.
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Mr. Andi note is student absent in last three year, that is January, February and March to 3 student that is Arlan, Bronto, and cery like at the table .
On the table can be written : Arlan Bronto Cery January February March 3 4 1 6 2 3 1 2 4
is a rectangular array of numbers, consists of rows and columns and is written using brackets or parentheses. The entries of a matrix are called elements of matrix . An element of a matrix is addressed by listing the row number and then column number M A T R I X
Matrix is generally notated using capital latter
2. The order of the matrix A matrix of A has m rows and n column is called as matrix of dimension on order m x n, and so notated of "A(mxn)". To more understand the definition of the element of a matrix.
The first column The second column The third column The column n-th The second row The first row The third row The row n-th
Example: Matrix A = The first row The second row The first column The second column The third column
The order matrix A is 2 x 3
4 is the second row and the first column
a row matrix Is a matrix that only has a row A = ( 1 3 5), and B = ( -1 0 4 7) The order matrix is and
a column matrix Is a matrix that only has a column
A matrix square A square matrix a matrix has the number of row of a matrix equals the number of its column
Example : rows 4, columns 4 A is matrix the order 4 A = Main diagonal
A = Upper Triangle Matrix is square matrix which all of the element under the diagonal is zero Upper Triangle Matrix
B = B is a lower triangle matrix is square matrix which all of the element upper the diagonal is zero Lower Triangle Matrix
C = Diagonal Matrix is square matrix that all of element is zero, except the element on the diagonal not all of them Diagonal Matrix:
I = I is matrix Identity that is diagonal matrix that elements at main diagonal value one Pay attention the following matrix
Transpose and Similarity of a Matrix
Transpose of a Matrix
Let A is a matrix whit dimension of (m x n). From the matrix of A we can formed a new matrix that obtained by following method:
a. Change the line of i th of matrix A to the row of
ith of new matrix
b. Change the row of j th of matrix A to the line of
jth of new matrix
The new matrix that resulted is called transpose from matrix of A symbolized with A' or From the above changess, the dimension of A' is (n x m)
Transpose matrix A A = IS A t =
Example :
let A = (aij) ang B = (bij) are two matrices with the same dimension. Matrix of A is callled equal with matrix of B id the element that located on the two matrices has the same value. 2. Similarity of two matrix
One located element with the same value One located element with the same value One located element with the same value One located element with the same value
Taking example A = and B = if A t = B, then determine the value x? Example 2:
Answer : A = = A t = B A t =
x + y = 1 x – y = 3 2x = 4 so x = 4 : 2 = 2
Algebraic Operation on Matrix
Addition and Subtraction of Matrix
Scale Multiplication with a Matrix
Matrix Multiplication with Matrix
Addition/Subtraction Two matrix can be summed/reduced if the order of the matrix are same and its statement in one position
Example 1: and B = A = A + B = + =
If A = , B = and C = hence(A + C) – (A + B) =…. Example 2:
(A + C) – (A + B) = A + C – A – B = C – B = = = Answer
Scale Multiplication With a Matrix Let k Є R and A is a matrix with dimension of m x n . Multiplication of real number k by matrix of A is a new matrix which is also has dimension of m x n that obtained by multiplying each element A by real number of k and notates kA
Matrix A = Determine matrix represented by 3A 3A = Example :1
Given Matrix of A = , B = and C = if A – 2B = 3C, So determine a + b ? Example 2 :
Matrix Multiplication with Matrix The Product Of Two Matrices A and B can be got when satisfies the relation A m x n = B p x q = AB m x q Equal
The number of column of matrix A should equal the number of rows of matrix B, the product, that is AB has order of m x q. when m is the number of rows of matrix A and q is the number of column of matrix B
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In this factoring worksheet, learners find the common term, use special products or use the diamond method to factor polynomials. Explanations and examples are provided. This three-page worksheet contains 45 problems. Answers are provided at the end of the document.
In this math study guide, students solve problems previously studied throughout the school year. Problems include, but are not limited to, volume, area, perimeter, square roots, binomials, trinomials, graphing, and consumer math. This eight-page worksheet contains 32 problems. Answers are provided on the last page of the worksheet.
Here's a fun way to explore and practice the concepts of multiplying monomials, multiplying binomials, and factoring polynomials. Included is the puzzle worksheet containing nine different sets. Don't worry, there's an answer key also.
Students factor numbers using Algebra Lab Gear. In this factoring quadratic equations instructional activity, students determine a number of different ways to factor a list of numbers. Students create a rectangle out of the pieces with no gaps and figure out what binomials could create such a rectangle for each problem. Students also create a chart in their groups that displays the attachments of each variable.
In this introduction to polynomial function worksheet, students classify, state the degree, find the intercepts, and evaluate polynomial functions. Six of the problems are true/false problems and thirty-eight and free-response problem solving problems.
In this perfect square worksheet, students factor perfect square trinomials and polynomials. They identify trinomials and polynomials that are perfect squares. This one-page worksheet contains 21 problems.
In this polynomial functions worksheet, students solve eight-six various problems concerned with the degree, graph, intercepts, end behavior, and transformations of polynomial functions. Some of the problems are algebraic work and some are graphical.
In this algebra worksheet, pupils solve a variety of problems. They find the greatest common factor of polynomials, factor polynomials and trinomials, and solve algebraic equations and real life story problems. Nine of the hundred problems require the interpretation of a story.
In this expressions and operations activity, 9th graders solve and complete 5 different multiple choice problems. First, they determine the complete factorization of various trinomials. Then, students find the equivalent equation to a completely factored trinomial.
In this perfect squares and factoring instructional activity, students solve and complete 26 different problems that include a number of polynomials. First, they determine whether each trinomial is a perfect square trinomial and factor if possible. Then, students factor each polynomial and write prime for those that are not.
In this completing the square activity, 11th graders solve and complete 24 different problems. First, they solve each equation by using the square root property. Then, students find the value of the variable that makes each trinomial a perfect square and write the trinomial as a perfect square.
A teacher guided lesson on perfect squares and factoring. They discuss perfect squares, observe the expansion steps for finding the product, and practice solving problems. They complete a worksheet on perfect squares and factoring.
In this expressions and operations worksheet, 9th graders solve and complete 10 different multiple choice problems that include various types of expressions. First, they determine the complete factorization of a given trinomial. Then, students find the solution for various equations shown. They also determine the solution set for a given equation.
Learners examine factoring polynomials. They observe a PowerPoint presentation as an introduction to factoring. Students factor binomials, trinomials and the difference of two perfect squares. Learners calculate problems after factoring them.
Students identify and describe polynomials and their elements, discuss simple parts that make up equations, inequalities, and exponents, take notes on definitions of terms and their types, including monomials, binomials, and trinomials, practice independently, and play flash card quiz game.
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It obviously requires single- and multi-variable calculus and linear algebra, but what else? And where do you suggest to get that background from?this isn't a duplicate because I'm for the math needed ...
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Pre-Algebra—Semester A
The number one place for all things number.
Course Description
Are you one of the many students out there that shivers at the thought of having to work with equations and graphs? What about rational and irrational numbers? Okay, enough with the scary words. Sometimes we think Stephen King doesn't have anything on the tales from middle and high school math classes.
What we're about to tell you might come as a surprise: Pre-Algebra isn't really that scary. Sure, there's something creepy about walking into a dark basement. But once the lights get flicked on, you just might find a surprise party waiting at the bottom of the stairs. (We wish!) Our goal is to turn on that light bulb and shed some insight—and hilarity—onto the darkness that is hiding the Pre-Algebra party.
In this course, we'll give you all the examples, practice problems, and projects you need to
learn all about the nuts and bolts of equations;
understand and use equations themselves;
and tackle graphs of lines, which happen to come from equations.
See how it all comes full circle?
P.S. Pre-Algebra is a two-semester course. You're
looking at Semester A, but you can check out Semester B here.
Technology Requirements
Microsoft Office, Google Docs, or another word processing program
A scanner (or access to one)
A camera (a camera phone is sufficient)
All other work can be done via the Shmoop website
Supported browsers:
IE 7+
Firefox 4+
Chrome 10+
Safari 4+
Opera 11+
Required Skills
This course is an accelerated course meant to be taken after 6th grade math and before algebra I. If you are looking for pre-algebra to be split into two years, check out our 7th and 8th grade math courses (coming soon).
No special technological skills are necessary for this course other than very basic computer literacy.
Course Breakdown
Unit 1. Expressions and Basic Operations
To be sure our math skills are in tiptop shape, we are going to start you out with some basic review. You may see some new and crazy things, like powers and roots, but other concepts will be old friends of yours, like addition and subtraction. Did we hear a sigh of relief or were those the baked beans you had earlier?
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Unit 2. Solving Equations
This unit is mainly about learning the rules that come along with expressions and equations, like the commutative and distributive properties. We'll also practice translating verbal sentences into written numeric and variable equations. No, that doesn't mean you can go around telling everyone you're bilingual.
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Unit 3. Multi-Step Equations and Inequalities in One Variable
This unit is all about solving multi-step equations and inequalities. It might feel a little shaky at first, but as long as you remember the basics of like terms and one-variable equations, you'll find yourself on solid ground again. And as long as you pay a little bit of attention, we'll make sure you don't get there face first.
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Unit 4. Radicals and Exponents
Here's where powers and roots will come back to haunt us. More like Casper the Friendly Ghost than the Exorcist, though. We'll go from prime numbers, GCFs, and factor trees to full-on radicals, exponents, and even negative powers (which—trust us—aren't nearly as evil as they sound).
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Unit 5. Rational and Irrational Numbers
Rational and irrational numbers are just types of numbers, like whole numbers and integers. Not only will we get to know these puppies, we'll learn how to use them in the real world. They might sound intimidating, but don't be fooled; they're all bark and no bite.
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Unit 6. Ratios and Proportions
In this unit, we'll get a really good flavor for how algebra and numbers mix together. Using fractions, units, and one-variable equations, we'll discuss ratios, proportions, rates, and we'll even do a fair amount of graphing on the coordinate plane. Who needs those dinky number lines, anyway?
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Unit 7. Equations in Two Variables
We'll learn all about how two-variable equations and inequalities can describe all sorts of different relationships—except your love-hate relationship with Lost. Those feelings are sort of inexplicable.
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Unit 8. Linear Equations and Functions
We start out this unit by learning about relations and functions, plotting ordered pairs, and graphing linear equations. Once we've got graphing down pat, we'll delve deep into the linear equations and their slope-intercept form. To top it all off, we'll tie these concepts together with a bit of modeling. Not that kind of modeling, so put that sequin dress away.
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