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Algebraic Equations Related Terms
Algebraic Equations Word Problems
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Many students find algebraic equations difficult. They feel overwhelmed with algebraic equations homework, tests and projects. And it is not always easy to find algebraic equations tutor who is both good and affordable. Now finding algebraic equations help is easy. For your algebraic equations homework, algebraic equations tests, algebraic equations projects, and algebraic equations tutoring needs, TuLyn is a one-stop solution. You can master hundreds of math topics by using TuLyn.
Our algebraic equations videos replace text-based tutorials and give you better step-by-step explanations of algebraic equations. Watch each video repeatedly until you understand how to approach algebraic equations problems and how to solve them.
Hundreds of video tutorials on algebraic equations make it easy for you to better understand the concept.
How to do better on algebraic equations: TuLyn makes algebraic equations easy.
Eighth Grade: Algebraic Equations Word Problems
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This resource from Nelson Thornes is the Teachers' Book which accompanies the Intermediate 2 Intermediate Higher Foundation 1 Students Book and is part of a series of books which were developed to deliver the three tier GCSE in Mathematics.
The chapter outline provided in the Teacher's Book begins with a list of the main learning outcomes for each chapter…
This resource from Nelson Thornes was for students following the Intermediate tier in mathematics Intermediate Tier in MathematicsA Year Eleven 5 contains some innovative content in most tasks with…
A Year Ten 4 contains some innovative content in most tasks with…
This resource from GAIM is a teacher assessment scheme, designed for use alongside any existing secondary mathematics programme of study, using open-ended tasks to encourage teaching and learning through practical problem solving and investigations.
One of the original aims of GAIM was to introduce continuous assessment into…
GAIM Activities are open-ended tasks where achievements in using and applying mathematics can be assessed alongside content. In the investigations students explore pure mathematics.
GAIM provides teachers with 80 Activities (40 Investigations and 40 Practical Problems) as a resource for teaching and assessment. These are open-ended…
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The process of receiving or learning new instructions and applying them practically in day to day life is something called as Education. The amount of knowledge gained varies from person to person. Education in today's fast moving world has become the key to survive. Education can either be through the process of learning or by work.
iTeach All is all about educating individuals in the field of those three subjects which are considered to be one of the most important domains in the field of education and they are - Mathematics, Physics & Chemistry. Another important vertical which is been included in the iTeach All is GMAT, which can be considered to be a part of Mathematics, as the vocabulary section of GMAT is not covered. Not just specific to students going to lower secondary levels or under graduate levels, but this application is also useful for those who are pursuing their post graduate studies or PhDs. Not just student specific, but this app is also recommended for professors, lecturers who can quickly grab useful information or any formula on the spot. Apart from the useful information and formulas, other important features are periodic table showing all the elements with temperature variation, GMAT tips & tricks which is very useful for applicant appearing for not only GMAT exam, but also any other competitive exam. It provides useful methods which are simple to understand, and helps you in solving complex equations with minimal efforts. It is a tutor to quicken your test taking abilities.
This application is designed by team Patron, and verified and checked by professionals who hold a Master's degree from University of Leicester, UK & San Jose State University, California.
Provides you with tips for how to solve complex equations and face competitive
examinations like GMAT, CAT, GRE, SAT.
Provides various methodologies and tips to tackle problems.
Easy drag & drop tool to find whatever you need with minimal fuss.
Periodic table with temperature variation shows how the elements behave under varying
temperatures.
Option of pinch and zoom out for better visibility of content.
Provides you with the option of copy & paste any part of the content with diagram.
The only complete educative app having both landscape and portrait view for better
understanding.
What makes iTeach All different from other apps is the amount of content, information and ease of access to any chapter with minimum efforts. It is true and a well know fact that there are good books available in the market, Sources from internet...etc, where information related to maths, physics and chemistry is easily available, but Patron provides a more mobile, flexible, easy and dynamic platform which helps anyone to revise as and when required
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Math 1351 Foundations of Mathematics II Information
LSC-CyFair Math Department
Catalog Description This is designed specifically for students who seek elementary and middle school teacher certification. Topics include concepts of geometry, probability, and statistics, as well as applications of the algebraic properties of real numbers to concepts of measurement with an emphasis on problem solving and critical thinking.
Course Learning Outcomes The student will: • Explore the geometric attributes of physical objects in order to classify and to form definitions. • Analyze spatial characteristics such as direction, orientation, and perspective. • Connect geometric ideas to numbers and measurement. • Use geometric models to solve problems. • Explore and understand measurement and estimation. • Analyze data and statistics. • Use probability with simple and complex experiments. • Understand surface area and volume through discovery.
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Books
Geometry & Topology
Learning geometry doesn't have to hurt. With a little bit of friendly guidance, it can even be fun! Geometry For Dummies,2nd Edition, helps you make friends with lines, angles, theorems devouring proofs with relishYou
Learning geometry doesn't have to hurt. With a little bit of friendly guidance, it can even be fun! Geometry For Dummies,2nd Edition, helps you make friends with lines, angles, theorems, understanding proofs like an expert You When you need to shape up, open up the included Geometry Workbook For Dummies, which contains over 290 pages with hundreds of practice problems featuring ample workspace to work out the problems. Each problem includes a step-by-step answer set to identify where you went wrong (or right). You'll be proving yourself proof-worthy in no time!
AUTHOR BIO: Mark Ryan owns and operates The Math Center in Chicago, a teaching and tutoring service for all math subjects as well as test preparation. He also wrote Geometry Workbook For Dummies.
Geometry is one of the oldest mathematical subjects in history. Unfortunately, few geometry study guides offer clear explanations, causing many people to get tripped up or lost when trying to solve a proof—even when they know the terms and concepts like the back of their hand. However, this problem can be fixed with practice and some strategies for slicing through all the mumbo-jumbo and getting right to the heart of the proof.
Geometry Workbook For Dummies ensures that practice makes perfect, especially when problems are presented without the stiff, formal style that you'd find in your math textbook. Written with a commonsense, street-smart approach, this guide gives you the step-by-step process to solve each proof, along with tips, shortcuts, and mnemonic devices to make sure the solutions stick. It also gives you plenty of room to work out your solutions, providing you with space to breathe and a clear head. This book provides you with the tools you need to solve all types of geometry problems, including:
Congruent triangles
Finding the area, angle, and size of quadrilaterals
Angle-arc theorems and formulas
Touching radii and tangents
Connecting radii and chords
Parallel, perpendicular, and intersecting lines and planes
Slope, distance, and midpoint formulas
Line and circle equations
Handling rotations, reflections, and other transformations
Packed with tons of strategies for solving proofs and a review of key concepts, Geometry Workbook For Dummies is the ultimate study aid for students, parents, and anyone with an interest in the field.
Features:Sizzling with Danica's trademark sass and style, her fourth book, Girls Get Curves, shows her readers how to feel confident, get in the driver's seat, and master the core concepts of high school geometry, including congruent triangles, quadrilaterals, circles, proofs, theorems, and more!
Combining reader favorites like personality quizzes, fun doodles, real-life testimonials from successful women, and stories about her own experiences with illuminating step-by-step math lessons, Girls Get Curves will make girls feel like Danica is their own personal tutor.
As hundreds of thousands of girls already know, Danica's irreverent, lighthearted approach opens the door to math success and higher scores, while also boosting their self-esteem in all areas of life. Girls Get Curves makes geometry understandable, relevant, and maybe even a little (gasp!) fun for girls.
This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications.
Lucid, well-written introduction to elementary geometry usually included in undergraduate and first-year graduate courses in mathematics. Topics include vector algebra in the plane, circles and coaxial systems, mappings of the Euclidean plane, similitudes, isometries, mappings of the intensive plane, much more. Includes over 500 exercises.
Anticipating the New York State Board of Regents' new examination in geometry, this brand-new classroom text presents a detailed review of all topics prescribed as part of the high school curriculum. Separate chapters analyze and explain: the language of geometry; parallel lines and polygons; congruent triangles and inequalities; special quadrilaterals and coordinates; similarity (including ratio and proportion, and proving products equal); right triangles and trigonometry; circles and angle measurement; transformation geometry; locus and coordinates; and working in space (an introduction to solid geometry). Each chapter includes practice exercises with answers provided at the back of the book.
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MATH301: Business Mathematics
Course Credits:
3
Course Hours Per Week:
12
Course Overview
This course provides a variety of applications-based math tools and concepts for the business professional. Teaching an effective foundation on topics that include product pricing, inventory valuation, depreciation methods, payroll, investments, costs of borrowing money, and accounting basics, the basics needed for initial analysis of financial situations in business are covered.
In the course project, Financial Plan: Using Business Math to Analyze the Financial Conditions of a Company, students apply the mathematical concepts that have been practiced throughout the course. A final report is submitted outlining different analyses, strategies for investments and borrowing, return on investments (ROI), and goals and action plans for the company.
Course Learning Objectives
Apply mathematical formulas to solve business-related math problems.
Use business-related tools to make effective financial decisions.
Analyze outcomes of spreadsheet results.
Special Requirements
Packaged with the textbook, Practical Business Math Procedures, students will receive a users guide and access code for a math practice program called ALEKS for Business Math. This web-based program requires Internet access, and it will be used in each module of this course. Please note that if you purchase the textbook from a source other than the JIU Bookstore, it may not include the access code you need to complete work in this course. Therefore, WE RECOMMEND THAT YOU PURCHASE THIS ITEM ONLY FROM THE JIU BOOKSTORE.
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realgebra, by definition is the transition from arithmetic to algebra. Miller/O'Neill/Hyde Prealgebra will introduce algebraic concepts early and repeat them as student would work through a Basic College Mathematics (or arithmetic) table of contents. Prealegbra is the ground work that's needed for developmental students to take the next step into a traditional algebra course.According to our market Julie and Molly's greatest strength is the ability to conceptualize algebraic concepts. The goal of this textbook will be to help student conceptua... MORElize the mathematics and it's relevancy in everything from their daily errands to the workplace.Prealgebra can be considered a derivative ofBasic College Mathematics.One new chapter introducing the variable and equations is needed. Each subsequent chapter is basic mathematics/arithmetic content with additional sections containing algebra incorporated throughout.
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A Level Further Mathematics
Introduction:
If you choose Further Maths you will be studying for two A-levels in Maths. This could mean that you spend half your time in College doing Maths. You have to really enjoy the subject. It is highly recommended if are considering studying a degree in Maths, Physics, Engineering or Economics at one of the best universities. Although you will get a whole A-level in a year it is not recommended for students who just want to get their Maths out of the way and do not intend to continue to the second year. It may be possible to pick up an AS in Further Maths in the second year which would involved just three of the six extra modules.
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Schaum's Outline of Beginning Finite Mathematics
thorough review-- with worked examples--of the fundamentals of linear equations and linear growth. Topics covered include games theory, descriptive statistics, normal distribution, probability, binomial distribution, and voting systems and apportionment.
The guides that help students study faster, learn better- and get top grades. This review of beginning calculus is updated to reflect the latest course scope and sequences, with expanded explanations ...
An ideal course text or supplement for the many underprepared students enrolled in the required freshman college math course, this revision of the highly successful outline (more than 348,000 copies ...
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We have been using both
2D and 3Dgraphics
to illustrate many mathematical concepts so far. In these pages we
will formalize some of the programming that was taken for granted.
In this chapter we will see several different types of 2D plots,
some special plots relevant to engineering, and the procedure for
creating basic animation
These pages emphasize generating
2D graphicsand plots using
the command lineand
function calls rather than use the interactive
plot editor for independence and exposure to programming.
Once again much more can be done than presented in these pages
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In Math C151141 skills will be prepared for Math C151.
Prerequisite MATH C142
In Math C151 students are expected to consistently interrelate the
multiple definitions of the trigonometric functions and their inverses;
determine the appropriate trigonometric ratio or law to apply to solve
problems with triangles; use the radian measure effectively in
conversions and it applying formulas to solve problems; analyze
trigonometric functions and their graphs using the concepts of
amplitude, period, phase and vertical shifts and apply these ideas to
real problems; recognize and verify or prove trigonometric identities;
analyze trigonometric equations to determine what combination of
algebra and identities will lead to a solution; apply trigonometry to
operations with complex numbers; solve problems and graph equations of
conic sections in rectangular and polar coordinate systems in two and
three dimensions; identify and solve problems using parametric
equations and vectors in the plane and in space. Students successfully
demonstrating these Math C142 skills will be prepared for Math C151.
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[b]MathTutor Differential Equations Vol. 1 First Order Equations[/b] mpeg4 .AVI | 10hours | Resolution: 720x540 | Audio: mp3 44100Hz 192 Kb/s | 6.90 GB [i]Genre: Elearning[/i][/center] Differential equations is used in all branches of engineering and science. In essence, once a student begins to study more complex problems, nature usually obeys a differential equation which means that the equation involves one or more derivatives of the unknown variable.
In other words, a differential equation involves the rate of change of a variable rather than the variable itself. The simplest example of this is F=ma. The "a" is acceleration which is the second derivative of the position of the object. Although differential equations may look simple to solve by just integration, they frequently require complex solution methods with many steps.
This 10 hour DVD course teaches how to solve first order differential equations using fully worked example problems. All intermediate steps are shown along with graphing methods and applications of differential equations in science and engineering.
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Please Note: Pricing and availability are subject to change without notice.
Math Pathways: Grades 6-8 from Sunburst Technology
Guided by NCTM, a teacher-guided, self-paced, curriculum and standard-based math learning tool that creates a learning environment in which students can explore, visualize, and appreciate mathematics.
Introduces and instructs students in major mathematic concepts utilizing a pathways methodology in which students are explained core mathematic concepts and then build on these concepts in their understanding of more complex mathematical problem solving. Guided by NCTM and designed as a teacher-guided, self-paced, curriculum and standard-based learning tool, Math Pathways creates a learning environment in which students can explore, visualize and appreciate mathematics. The unique combination of 3D animation and exercises will ensure all math students acquire necessary conceptual understanding and computational skills to achieve high school standards.
With Math Pathways students will:
Learn concepts and then build upon these concepts using various pathways that converge and build upon each other such that retention will be far greater than the traditional rote memorization method of teaching mathematics
Access an online tool which allows them to self guide students through middle and high school major mathematical concepts
Illustrate math concepts and their applicability to "real world" situations
System Requirements
Windows Platform
Pentium
Win 98/Win ME/Win 2000/Win NT/Win XP
32MB
Macintosh Platform
PowerMac
Mac 8.6/Mac 9.1/Mac OSX
32MB
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Textbook lessons are divided into three sections. The first section is "power-up practice," which covers basic fact and mental math exercises which improve speed, accuracy, the ability to do mental math, and the ability to solve complicated problems. The second part of the lesson is the "New Concept," which introduces a new math concept through examples, and provides a chance for students to solve similar problems. Thirdly, the "Written Practice" section reviews previously taught concepts. One "Investigation" per session is included; "Investigations" are variations of the daily lesson and often involve activities that take up an entire class.
The included Power Up Workbook provides consumable pages for students to complete the Power Up exercises from the textbook, including the Facts Practice, Jump Start, Mental Math, and Problem Solving sections. The textbook may refer students to problems within this Power Up workbook, or the text may contain necessary problems and instructions (such as the mental math problems), which students will need to complete the exercises in this workbook.
The Solutions Manual arranges answers by section and lesson, and includes complete step-by-step solutions to the Lesson Practice, Written Practice, and Early Finishers questions, as well as the questions and practice items in the Investigations. It does not contain the answers to the Power-Up Workbook, which are currently unavailable.
The Homeschool Testing Book features reproducible cumulative tests which are available after every five lessons after lesson 10. Tests are designed to let students learn and practice concepts before being tested, helping them build confidence. Tests, a testing schedule, test answer forms, test analysis form, and test solutions are included. The three optional Test Solution Answer Forms provide the appropriate workspace for students to "show their work." The answer key shows the final solution only, not the steps taken to arrive at the answer.
Big Fan of Saxon Math
Date:March 20, 2013
ScottI like the Saxon Math concept of incremental development (the student revisits concepts already learned). It seems to keep the knowledge fresh. Occasionally, we will enhance the curriculum with a little deeper dive into the subject matter... but perhaps I do that to make myself feel useful! If I had a fault with the program it would be that each new year starts a little slow (I'm sure the authors assume the student has taken the summer off).
Share this review:
+1point
1of1voted this as helpful.
Review 2 for Saxon Math Intermediate 3 Complete Homeschool Kit
Overall Rating:
5out of5
Date:July 6, 2012
Judy R
Location:Baytown TX
Age:Over 65
Gender:female
Quality:
4out of5
Value:
5out of5
Meets Expectations:
5out of5
I have been a long-term Saxon math user. My daughter who is now 30 years old used it, and now I am using the same series for my grandchildren, ages 7 and 10. I love the way the Saxon series has lots of repetition, and the way that material is presented in small bits and pieces. Both of my grandchildren love math and say it's so easy to do the "hard stuff" their friends have to do in public school.
Share this review:
+1point
1of1voted this as helpful.
Review 3 for Saxon Math Intermediate 3 Complete Homeschool Kit
Overall Rating:
5out of5
Perfect for those looking for a lot of repetition.
Date:September 6, 2011
Xgraver
Location:Lancaster have used Saxon K and 1st grade. I was looking for a curriculum that would just focus on the basics before moving on. I found that with Saxon Intermediate 3. The earlier versions were just too childish for my kids at this point. We did not require the use of manipulatives anymore and Inter. 3 provided that.
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Standard Deviants Light Speed is a comprehensive video course designed to teach core
curriculum subjects. Our rapid-fire approach keeps students engaged while our cast of young
actors, humorous skits, mnemonics, and on-screen graphics present step-by-step examples to
teach students the essentials to master a subject.
Each module includes a video program plus a Digital Workbook that follows along with
the program to reinforce the video material. The subject matter correlates directly to state
standards and is produced and designed by an academic team of professors, students, actors,
comedians, and teachers. Fast forward to success with this personal tutor in a box!
LIGHT SPEED ALGEBRA
This program introduces students to our friend, Algebra, the powerful super hero that is armed
with the knowledge to find out fascinating things about place, people and real-life situations.
This series concentrates on the essentials of algebra plus provides a thorough breakdown of
difficult concepts using step-by-step explanations and visual examples.
THE POWERS AND FUNCTIONS OF ALGEBRA
For many students the study of algebra is similar to learning a new language, it can be very
confusing and daunting. Our cast of young actors define those difficult algebra terms and
use on-screen graphics to present step-by-step examples. This program puts the FUN into
functions and other difficult algebra topics and is a must see for any student who is starting
algebra or needs a refresher.
SOLVING, SIMPLIFYING AND SLOPE Join us as we deep dive into some of the more advanced concepts and principles of Algebra -
Equalities, Inequalities, and Graphing these equations. We unlock the world of simplifying the
complex and use real-life scenarios to make algebra relevant!
LINEAR EQUATIONS AND POLYNOMIALS Less then, greater than, monomial, polynomials, oh my! The more we learn about Algebra, the
more complex the equations and vocabulary become. Have no fear; Light Speed is here to
make these equations, terms and concepts digestible.
ESSENTIAL QUADRATICS
Interested in finding out how high a baseball can fly? Or how fast a record player will drop
from a window? The quadratic formula can be used to figure out projectile motion – meaning,
how high and how far a moving object travels. This program will present the equation, how it
is graphed and how it can be solved. You'll soon be a quadratic expert!
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Jobs
Mathematics AS
Introduction to course
Mathematics is a popular choice with students; typically we have two hundred students studying AS Mathematics in the first year with over thirty of these opting to study Further Maths AS as well.
Course Details
You must enjoy Mathematics to consider it for an A Level subject; you should get a kick out of solving problems correctly and enjoy a challenge. In the first year, students study two Pure Maths modules (C1, C2) and Decision Maths (D1).
In the second year, you will study C3, C4 and Mechanics (M1). We follow the AQA syllabus.
Entry Requirements
In addition to the general entry requirements you are also required to have the following:
Where the course lead
Mathematics is essential if you want to study for a Mathematics degree and is often necessary for Physics, Engineering or Computing. It can also provide useful support for further studies in Biology, Chemistry, Finance, Business, Economics and Social Sciences.
An A and AS Level in Mathematics demonstrates that you have a level of numerical and problem solving skills which are well above average, making you particularly valued by employers and higher education establishments.
This course combines well with
Students doing A Level Maths study a wide variety of other subjects - the most popular being Economics and the Sciences. However, students studying Humanities and the Social Sciences are also well represented in Maths classes.
Course Assessment
3 x 1 and a half hour exams at AS
3 x 1 and a half hour exams at A2
Links
mathscareers.org (an excellent site for researching careers in maths and lots of exciting examples of how maths can be used in the real world)
What help is available?
Clearly most of your learning will take place inside the classroom, but as well as excellent teaching we also provide
Timetabled workshops for students who need extra help
Drop-in workshops for all students
Up-to-date textbooks
A range of resources, web-links and all assessment materials available on the college intra-net (moodle)
A good range of additional textbooks in the library
Revision, study guides and graphical calculators for sale
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No other current books deal with this subject, and the author is a leading authority in the field of computer arithmetic. The text introduces the Conventional Radix Number System and the Signed-Digit Number System, as well as Residue Number System and Logarithmic Number System. This book serves as an essential, up-to-date guide for students of electrical engineering and computer and mathematical sciences, as well as practicing engineers and computer scientists involved in the design, application, and development of computer arithmetic units. less
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One reason for including certain material in the education of "all mathematicians" ... my example is complex analysis. You may end up teaching at a small college, where you teach all undergraduate math courses. Including complex analysis, since engineering and physics want their students learn it. And it is desirable that the instructor know more than just what is in the textbook.
Many times I have heard complaints of this kind... "My research is in graph theory so I will never need to know complex analysis, why do I have to waste time learning it?"
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MAT1102 Algebra & Calculus I
Review question
Calculus
What is the difference between the derivative function and the derivative at a point?
Given f(t), how can the first derivative f(t) be written in Leibniz notation? How about the second deriv
MAT1102 Algebra & Calculus I
Review question
Calculus
What is the difference between the derivative function and the derivative at a point?
Given f(t), how can the first derivative f'(t) be written in Leibniz notation? How about the second deriRelating coalgebraic notions of bisimulation
with applications to name-passing process calculi (extended abstract)
Sam Staton
Computer Laboratory, University of Cambridge
Abstract. A labelled transition system can be understood as a coalgebra for a
Math 486 Exam 3 Review Sheet
16 April 2008 This document is a preliminary review sheet. Since it is prepared well before the nal exam, it is possible that not all of the material on this review sheet will be covered on the nal exam. The nal exam is c
Chapter 2
Equational Logic
2.1
2.1.1
Syntax
Terms and Term Algebras
The natural logic of algebra is equational logic, whose propositions are universally quantied identities between terms built up from variables with operation symbols. For example
Mathematics TExES Competencies 8-12 DOMAIN INUMBER CONCEPTS Competency 001 The teacher understands the real number system and its structure, operations, algorithms, and representations. The beginning teacher: Understands the concepts of place value,
Abstract algebra
Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term abstract algebra is used to distinguish the field from "elementary algebra" or "high school alge
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An Introduction to Differential Equations and Linear accessible guide offers a thorough introduction to the basics of differential equations and linear algebra. Expertly integrating the two topics, it explains concepts clearly and logically -without sacrificing level or rigor and supports material with a vast array of problems of varying levels for readers to choose from.
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Curves in the plane and curves in space. Orientable surfaces. Introductory vector calculus.
The course also connects to mathematical education by treating for example oral and written presentations of mathematics and the use of calculators and mathematical software.
Eligibility
SF1623 Mathematics I for CL.
Literature
Andersson Lennart m.fl. : Linjär algebra med geometri.
Persson&Böiers/Analys i flera variabler.
LTH/Övningar i analys i flera variabler.
Examination
ANN1
- Assignment,
,
grade scale:
A, B, C, D, E, FX, F
TEN1
- Examination,
12.0 credits,
grade scale:
A, B, C, D, E, FX, F
TEN2
- Examination,
7.5 credits,
grade scale:
A, B, C, D, E, FX, F
TEN3
- Examination,
4.5 credits,
grade scale:
A, B, C, D, E, FX, F
Requirements for final grade
Normally the examination consists of TEN1 (12 hp), continuous examination combined with a final written exam, and ANN1 (0 hp, compulsory) on a selected topic in the didactics of mathematics. TEN1 may under certain circumstances be replaced by TEN2 (7,5 hp) and TEN3 (4,5 hp). TEN2 corresponds to the following parts of the course: linear algebra, curves in the plane and curves in space and differential calculus and extreme value problems for functions of several variables. TEN3 corresponds to series, multiple integrals and introductory vector calculus.
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MATH 093: Pre Algebra
General Information
Transfer Information: Courses with numbers below 100: This course is a developmental course that does not apply toward a WNC degree or honors designation and normally does not transfer to a university. Please see a counselor for more information.
Prerequisites and Recommended Courses
Course Outline
I: Catalog Course Description
Prepares students for MATH 95. Helps students who have experienced difficulties with math to get an introduction to the language and concepts of algebra. Provides a transition from self-paced, basic math to the quick pace required in MATH 95.
II: Course Objectives
Upon completion of this course, successful students should be able to work with the following:
Prime and composite numbers
Properties of and operations on signed numbers
Linear equations
Ratios, Rates and Proportions
Solving applied problems
Rectangular Coordinate System
Graphing Linear equations
Square Roots and Pythagorean Theorem
III: Course Linkage
Linkage of course to educational program mission and at least one educational program outcome.
Although Math 93 does not fulfill any general education or degree program requirements, it will assist students toward developing appropriate college-level mathematical skills and problem solving skills.
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Math Study Skills outlines good study habits and provides students with study strategies and tips to improve in areas such as time management, organization, and test-taking skills. With a friendly and relatable voice, Alan Bass addresses the misgivings and challenges many students face in a math cla...
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Topics such as geometry, computing,
algebra, number theory, history of mathematics, logic, probability,
statistics, modeling and problem solving intended to give students
insight into what mathematics is, what it attempts to accomplish and how
mathematicians think.
Sets, logic, numeration systems, number
theory, probability and statistics, measurement, geometry and an
introduction to computers. This course will fulfill the requirements for
licensure of prospective early childhood and middle school teachers, as
well as provide a general introduction to mathematics for students in
other majors. Prerequisite for MATH 108: MATH 107.
MATH 135. Elementary Functions. 4 credits.
Algebraic, exponential, logarithmic and
trigonometric functions; matrices and matrix solutions to systems of
linear equations; vectors. Not open to students who have previously
earned credit in MATH 125, 145, 155, 156, 205 or 235, except with the
consent of the department head.
MATH 155. College Algebra. 3
credits.
Polynomial, rational, exponential and
logarithmic functions and applications, systems of equations and
inequalities, sequences. Prerequisite: Demonstration of proficiency
in algebra at an intermediate level. A test is required to determine
placement in MATH 155 or MATH156. Not open to students who have
previously earned credit in MATH 125, 135, 145, 156, 205, 220 or 235.
MATH 156. College Algebra. 3
credits.
Covers same topics as MATH 155. MATH 156
will meet five times a week for students requiring more instructional
time. Prerequisite: Demonstration of proficiency in algebra at an
intermediate level. A test is required to determine placement in MATH
155 or 156. Not open to students who have previously earned credit in
MATH 125, 135, 145, 155, 205, 220 or MATH 235.
MATH 167. Topics in Mathematics. 1-3 credits.
Topics or projects in mathematics which are
of interest to the lower-division student. May be repeated for credit
when course content changes. Topics or projects selected may dictate
prerequisites. Students should consult the instructor prior to enrolling
for this course.
*MATH 205. Introductory Calculus I. 3
credits.
Topics from differential calculus with
applications to the social, behavioral or life sciences and business or
management. Prerequisite: Demonstration of strong preparation in
algebra. Not open to mathematics or physics majors or to students
who have already earned credit in MATH 235. Not recommended for
chemistry majors.
Topics from integral calculus with
applications to the social, behavioral or life sciences and business or
management. Prerequisite:MATH
205. Not open to mathematics or physics majors or to students who have
already earned credit in MATH 236. Not recommended for chemistry majors.
*MATH 220. Elementary Statistics. 3 credits.
Descriptive statistics, frequency
distributions, sampling, estimation and testing of hypotheses,
regression, correlation and an introduction to statistical analysis
using computers. Prerequisite: Demonstration of strong preparation in
algebra. Not open to majors in mathematics.
MATH 235*-236. Calculus I-II. 4 credits each
semester.
Differential and integral calculus of
functions of one variable. Sequences and infinite series. Prerequisite
for MATH 235: MATH 135 or equivalent. Prerequisite for MATH 236: MATH
235 with grade of "C" or better.
Programming in a high-level computer
language. Applications of numerical algorithms to problems basic to
areas such as mathematics, the sciences and economics and finance. Prerequisite
or corequisite: MATH 206 or MATH 236. This course is not open to
students who have previously earned credit in MATH/CS 448.
MATH/PHYS 265. Introduction to Fluid Mechanics. 4
credits.
Introduces the student to the application
of vector calculus to the description of fluids. The Euler equation,
viscosity and the Navier-Stokes equation will be covered.
Prerequisites: MATH 237 and PHYS 260.
MATH 280.SAS
Programming and Data Management.3
credits.
Use of statistical software to manage,
process and analyze data.Writing
of statistical programs to perform simulation experiments. Prerequisites:
MATH 220 or MATH 318.
MATH 285. Data Analysis. 4
credits.
Topics include experimental and survey
design, distributions, variation, chance, sampling variation, computer
simulation, bootstrapping, estimation and hypothesis testing using real
data generated from classroom experiments and large databases. Prerequisite:
MATH 206 or MATH 236 or permission of instructor. Not open to students
who have already earned credit in MATH 220 or MATH 318.
Introduction to basic concepts in
statistics with applications of statistical techniques including
estimation, test of hypothesis, analysis of variance and topics in
experimental design. Prerequisite: MATH 220, MATH 318 or equivalent.
MATH 322. Applied Linear Regression. 3
credits.
Introduction to basic concepts and methods
in regression analysis and the application of these models to real-life
situations. Prerequisite: MATH 220, MATH 318 or equivalent.
Exact inference for population proportions,
comparison of population proportions for independent and dependent
samples, two and three-way contingency tables, Chi-square tests of
independence and homogeneity, Chi-square goodness-of-fit tests and
Poisson and logistic regression. Prerequisites:MATH 220 or MATH 318.
Development of techniques for obtaining,
analyzing and graphing solutions to differential equations, with
emphasis on first and second order equations. Prerequisite: MATH 237.
MATH 337. Methods of Applied Calculus. 4
credits.
Laplace transforms, power series and their
application to differential equations. Vector differential and integral
calculus; parametric curves; coordinate systems; line, surface and
volume integrals; and gradient, divergence and curl including the
theorems of Green, Stokes and Gauss. Prerequisite: MATH 237 and MATH
238.
MATH 340. Mathematical Modeling I –
Optimization. 3 credits.
Linear and nonlinear optimization with an
emphasis on applications in the sciences, economics and social sciences.
Techniques studied include the simplex, Newton and Lagrange methods and
Kuhn-Tucker theory. Software packages will be used to implement these
methods. Prerequisite: MATH 248.
MATH 341. Mathematical Modeling II – Dynamical
Systems.3
credits.
Discrete and continuous dynamical systems
with an emphasis on applications in the sciences, economics and social
sciences. Topics include stability, chaos, phase portraits, strange
attractors and fractals. Software packages will be used to model the
dynamical systems. Prerequisite: MATH 340; or MATH 238 and MATH 248.
Applications of computer models to the
understanding of both compressible and incompressible fluid flows. Prerequisites:
MATH 248, either MATH 238 or MATH 336, MATH/PHYS 265 and PHYS 340.
MATH/PHYS 366E. Computational Solid Mechanics. 3
credits.
Development and application of mathematical
models and computer simulations to investigate problems in solid
mechanics, withemphasis on
numerical solution of associated boundary valueproblems.Prerequisites:
MATH/PHYS 266, MATH 238 and MATH 248, or consent of instructor.
An overview of the role of mathematical
concepts in financial applications. Topics include continuous time
finance, optimization, numerical analysis and applications in asset
pricing. Prerequisite: MATH 237 and FIN 380.
MATH/FIN 405. Securities Pricing. 3 credits.
A quantitative treatment of the theory and
method of financial securities pricing to include an examination of
closed form pricing models such as the Black-Scholes and its various
derivatives as well as numerical solution techniques such as binomial
methods. Prerequisite: MATH/FIN 395.
Development and use of probability and
statistics for strategic decision making with applications. Topics
include decision flow diagrams, analysis of risk and risk aversion,
utility theory, Bayesian statistical methods, the economics of sampling,
sensitivity analysis and collective decision making. Prerequisite:
MATH 318.
Sampling theory and distributions, theory
and applications of estimation and hypothesis testing, regression and
correlation and analysis of variance. Prerequisite: MATH 426.
MATH 429. Research Project in Statistics. 1-3
credits.
Experience in the design, data collection
and analysis for a survey or experiment.MATH 429 should be taken concurrently with one of the following
statistics courses:MATH
321, 322, 324, 325, 327. Corequisite:One of MATH 321, MATH 322, MATH 324, MATH 325, or MATH 327.Prerequisite: Six hours of statistics courses at the
300 or 400 level.
MATH 430-431. Abstract Algebra. 3 credits each semester.
An introduction to groups, rings and
fields. Prerequisite for MATH 430: MATH 238 or MATH 300 and one of
MATH 310, 312 or 315 or consent of the instructor. Prerequisite for MATH
431: MATH 430.
Analysis of qualitative properties and
dynamics of linear and non-linear ordinary differential equations,
including topics such as existence, uniqueness, phase portraits,
stability and chaos, with applications to the sciences. Prerequisites:
MATH 238; and either MATH 310, 312, 315, 387 or consent of instructor.
MATH/FIN 465. Seminar in Actuarial Science I. 3
credits.
Theory and application of contingency
mathematics in the areas of life and health insurance and of annuities
from both a probabilistic and deterministic approach. This class,
together with MATH/FIN 466, helps students prepare for the professional
actuarial examinations. Offered fall, odd-numbered years. Prerequisite:
MATH/FIN 395 or consent of the instructor. Prerequisite or corequisite:
MATH 426.
MATH/FIN 466. Seminar in Actuarial Science II.
3 credits.
A continuation of MATH/FIN 465. Additional
coverage of contingency mathematics in the areas of life and health
insurance, annuities, pensions and risk theory from both probabilistic
and deterministic approaches. The two-course sequence helps to prepare
the student for the professional actuarial examinations. Prerequisite:
MATH/FIN 465. Prerequisite or corequisite: MATH 427.
MATH 467. Selected Topics in Mathematics.1-3 credits each semester.
Topics in advanced mathematics which are of
special interest to the student but not covered in the regularly offered
courses. Offered only with approval of the department head; may be
repeated for credit when course content changes.
MATH 475. Fundamental Concepts of Geometry.
3 credits.
Origin and development of Euclidean and
other geometries including axiomatic systems, mathematical proof and
special topics from incidence geometry. Prerequisite: MATH 310, MATH
312 or MATH 315 or consent of the instructor.
MATH 483. Selected Topics in Applied Mathematics.
3 credits.
Topics in applied mathematics not covered
in the regularly offered courses. Offered only with the approval of the
department head; may be repeated for credit when course content changes.
Prerequisites: Junior standing and consent of the instructor.
MATH 485. Selected Topics in Statistics. 1-3
credits.
Topics in statistics which are of interest
but not otherwise covered in the regular statistics offerings of the
department. Offered only with approval of the department head; may be
repeated for credit when course content changes. Prerequisites:
Junior standing and consent of the instructor.
MATH 497-498. Independent Study. 1-3 credits each semester.
Independent study in mathematics under
faculty supervision. Offered only with consent of the department head.
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Elementary Algebra
9780495105718
ISBN:
0495105716
Pub Date: 2006 Publisher: Thomson Learning
Summary: Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; use the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundam...ental problem solving skills necessary for future mathematics courses in an easy-to-read format. The Eighth Edition of ELEMENTARY ALGEBRA includes new and updated problems, revised content based on reviewer feedback and a new function in iLrn. This enhanced iLrn homework functionality was designed specifically for Kaufmann/Schwitters' users. Textbook-specific practice problems have been added to iLrn to provide additional, algorithmically-generated practice problems, along with useful support and assistance to solve the problems for students [
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The program helped my son do well on an exam. It helped refresh my memory of a lot of things I forgot. Jessica Simpson, UT
You have made me a believer. Your support and quick response is key. Thank you again for your professional support, easy to talk to, understanding, and patient. Joanne Ball, TX
Just watching my students, one after another, easily grasp these higher mathematical concepts and really, truly understand what they are doing, makes Algebra Buster worth the price of admission. Furthermore, for the gains, the price is tremendously inexpensive! Anthony Washington, MO09-27:
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Functions
Functions teaches students how to properly use function notation and how to answer questions related to them, such as how to find domain and range, how to define a function, and how to interpret word problems with direct and indirect variation.
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If you would rather leave me a VOICE MAIL - call the school at 541-445-2131 and my extension is x209. This will record a message on my room phone that I will have access to.
Of course you can always leave a message with Ms. Dever in office as well.
GRADES
Gradebooks are available through the PASS System. All you need to access this is your child's student id number and a password which you can receive from Jeanine Dever in the main office. This is an excellent way to monitor your child's grades on a regular basis.
Look here to see class assignments. This will provide families and students access to what was assigned in class in case they were absent or missed class. It will also provide a way for parents to confirm whether or not students have HW.
CURRENT WEEK'S INFORMATION
ALGEBRA I
ALGEBRA II
MATH 8
MATH 7
ENRICHMENT grade 8
GEOMETRY
Monday
Tuesday
Wednesday
Thursday
LAST WEEK'S INFORMATION
ALGEBRA I
ALGEBRA II
MATH 8
MATH 7
ENRICHMENT grade 8
GEOMETRY
MONDAY
TUESDAY
WEDNESDAY
THURSDAY
Course Statements
MATH 7: All the expected algebraic topics are covered in this text. Patterns, relations, and functions are presented early in the text and are reviewed and practiced throughout the year. Order of operations are applied to whole numbers, integers, rational numbers, and exponents. Students build on their understanding of variables and expressions and extend them to equations and inequalities. Students also analyze patterns and functions leading to graphing on the coordinate plane.
MATH 8: Similar to Course 2 however the development of algebraic thinking progresses from Course 1 to Course 3, building a solid foundation for students to have confidence and success in Algebra I.
ALGEBRA I: Saxon Algebra 1 covers advanced topics such as arithmetic of and evaluation of expressions involving signed numbers exponents and roots. Students learn properties of the real numbers; absolute value and equations or inequalities involving absolute value; unit conversions; solution of equations in one unknown and solution of simultaneous equations; polynomials and rational expressions; word problems requiring algebra; Pythagorean theorem; functions and functional notation; solution of quadratic equations; and much, much more.
GEOMETRY: Saxon Geometry books teach postulates and theorems and two column proofs. They also teach triangle congruence, surface area and volume, vector addition, slopes and equations of lines. With topics like these, Saxon Geometry books cover all the ground of a traditional high school geometry course, with some additional topics thrown in to connect with real life applications as well as Algebra review.
ALGEBRA II: Saxon Algebra 2 topics covered include: graphical solution to simultaneous equations; roots of quadratic equations, even including complex roots; inequalities and systems of inequalities; logarithms and antilogarithms; exponential equations; basic trigonometric functions; vectors; polar and rectangular coordinate systems, and so much more! There are also many different types of word problems requiring algebra in their solution, and real world applications in areas such as physics and chemistry are discussed.
Saxon Algebra 2 books are rather unique in that they not only cover second year algebra, but also a good deal of geometry, equaling about a semester's work of informal geometry, including proof outlines. There is also treatment of set theory and probability and statistics.
Ms. Windus' Schedule
I am excited to return to Camas Valley and looking forward to continue to assist students as they strengthen and learn mathematics.
This year I will teach the following:
period 1 - Algebra I
period 2 - Algebra II
period 3 - Math 8
period 4 - Math 7
period 5 - 8th grade Math Enrichment
period 6 - Geometry
Extra Help is available for students who need more assistance. A schedule will be posted in the classroom(and here) once developed.
STUDENT SUPPLIES
The following explains materials students should have while in math class and those that should be made available at home. (This is a list in progress - check for updates)
IN CLASS
3 ring binder - 1"-1.5"
lined notebook paper - can be a spiral notebook to be carried in binder
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Directories
Bardzell, Michael
MATH 201-450 475 Prerequisite: MATH 202 or 210. Three hours per week.
MATH 575 Three hours per week.
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Format:Students willlearn in a classroom that is focused on higher-order thinking skills, problem solving, substantive conversation (orally and written), and real world contexts.The goal of this course is to allow students to work individually and in groups to build on their content knowledge through conversation with each other in order to show their understanding of algebra, geometry, and statistics by completing performance tasks.
Topics: This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric and inverse trigonometric functions; basic trigonometric identities and the laws of sines and cosines; sequences and series; polar and parametric equations; vectors; the central limit theorem and confidence intervals. (Prerequisite: Accelerated Mathematics 2 or Mathematics 3.) Instruction and assessment should include the appropriate use of manipulatives and technology. Topics should be represented in multiple ways, such as concrete/pictorial, verbal/written, numeric/data-based, graphical, and symbolic. Concepts should be introduced and used, where appropriate, in the context of realistic phenomena.
Supplies:It would be helpful for students to have a graphing calculator (TI-84 Plus, TI-89, or TI-Nspire) for this class; however, graphing calculators will be provided for in-class use.Students will need pencils, pens, paper, and graph paper.I also suggest you keep class notes in a notebook.
Notebook: The student will keep an organized notebook containing sections labeled for each unit. The Post-It tabs will be used to separate vocabulary and tasks within a unit. The notebook may be graded and count as a quiz grade.
Extra Help: I am always willing to give students extra help. Students will need to make an appointment so that I can give him/her a pass and make arrangements for the day that he/she is coming in for help.
üNote: Students are expected to be prepared for each tutoring session by having their notebook and specific problems/questions to address.
Classroom Rules:
All school rules, including dress code, cell phones, I-pods, and tardies will be enforced. These are 5 P's that will make you a success in my class:
Violations of school rules will be dealt with as outlined in the Code of Conduct handbooks. This includes enforcement of the absence and tardy policies. Within the classroom, consequences of rule-breaking can include:change of seating, loss of privileges, student-teacher discussion, call to parents, grade penalties, detention, referral to administration, and other appropriate consequences as merited by the situation.
Procedures:
1. Sharpen pencils upon entering the classroom and answer essential questions from the screen.Essential questions should be answered within the first ten minutes of class.
2. Every day you must have the necessary supplies with you when you get to class. Please bring your textbook, notebook, pencils, and paper.
3. You will be placed in groups randomly.You will work individually in groups to complete daily assignments.Conversations should be limited to your group about academics only, which means no conversations about personal adventures.You are to work the entire class period in your groups.At the end of each block, we will share information as a class.I will answer any additional questions you may have at that time. There may be problems that are unusually difficult that I have not prepared you for. In this case, you need to write down what you know, what you have tried, and the questions you will ask me that will get you going on the problem. If you have shown this effort, and written this information on your assignment, I will grade it according to your effort and your explanation.
Label the top right hand corner of each assignment:
First and Last Name
Acc Math III and Block # A or B
GPS Standard Addressed
4. Please use the restroom, chat with teachers, and friends before class begins. You are responsible for any information and/or materials you miss while you are out of the room. No passes will be issued in class unless it is an emergency.Exceptions will only be given with a doctor's note.
5. You may not use any portable music, game, telephone or TV devices in class.You must remain in dress code at all times.
6. Remember that many of life's problems can be prevented or solved with a little communication. If you are having difficulty in class, please come talk to me. We can usually find a solution.
7. No food or drink
***Keep this in mind: if you ever feel that you might be falling behind, you probably are!Please get extra help immediately.The longer you wait to get help, the more difficult it will be to get back on track.
Grading: Grades will be given as follows:
A………90-100
B………80-89
C………70-79
F………Below 70
Your grade will be based on the following percentages:
Notebook………........…………....15%
Classwor /Homework Projects…….20%
Tests………..…………………….25%
Quizzes…………………...…….....20%
Performance Assessments ………...20%
EOCT………………………….....15%
Your grade at the end of First Semester will be your Mid-semester grade.Your final grade will be given at the end of Second Semester.
Personal Note:
I will do my best to make this year a pleasurable learning experience.I ask that you apply yourself, be engaged, and ask questions when you need answers.I will gladly give you special assistance if you need it.
Special Note to Parents:
I want your child to succeed.I will do the best I can to make that happen.You can help by ensuring that your child is in class every day with appropriate materials and that your child has completed all homework. I would greatly appreciate having your email address, if you have one.Email is the best way to communicate with me and the easiest for me to use if I need to get in touch with you.My email is at the top of this syllabus, and there is a direct link to my email on the website.
Tear here
I acknowledge that I have received and read the syllabus.I understand that the schedule is subject to change, but I am responsible for all information.
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Statistics 2 at Berkeley is an introductory class taken by about 1000 students each year. Stat2.2x is the second of three five-week courses that make up Stat2x, the online equivalent of Berkeley's Stat 2.
A degree in General Mathematics is designed to equip you with the skills necessary to be a professional problem-solver, as a mathematician is not defined by his or her knowledge of laws and theorems, but by his or her critical thinking and reasoning skills.
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New GCSE Maths - Algebra Revision App: for Android
Part of the
New GCSE Maths series
Digital download • 978-0-00-745852-3 • Oct
2011
£1.49
Availability:
Temporarily unavailable
Revise with Collins New GCSE Maths anywhere with Collins Revision Algebra.This interactive maths resource will identify trouble areas and suggest further studies, and is full of video tutorials and real-life examples to help you revise and practise, progress and get top exam results.
With Collins Revision Algebra you can now hone your mathematics skills wherever you are.
• Be inspired by interactive animations, exciting video clips of students teaching a problem, real-life examples of maths at work, worked exam questions, tutorials and more • Choose precisely which topics to revise and practise with material corresponding to the Collins New GCSE Maths scheme • Test yourself with interactive assessment questions that identify trouble areas and suggest relevant further revision • Zoom into graphs and diagrams for a closer look and better understanding
Collect all four Apps covering all four GCSE maths strands, for a total of 900 practice questions, 300 assessment questions and 130 video clips!
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MATH 333 Introduction to Abstract Mathematics
An introduction to mathematical rigor and proof encountered in advanced mathematics. Topics include logic, sets, elementary number theory, relations, functions, limits, cardinality, the complex number system
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...
More About
This Book
think about the discipline. Perfect as a supplemental guide for all geometry courses,Geometry for the Clueless includes:
Expert tips and techniques for solving difficult problems
Non-technical language
A unique way of explaining complex concepts to make learning geometry less daunting
STUDENT TESTED AND APPROVED!
Do you suffer from math anxiety? Do theorems,figures,and angles leave your head spinning? If so,you are like hundereds of thousands of other sutdents who face mathespecially,geometrywith fear.
Luckily,there is a cure: Bob Miller's Clueless series!
Like the teacher you always wished you had (but never thought existed),Bob Miller brings a combinatin of knowledge,empathy,and fun to the often-troubling subject of geometry. He breaks down the learning process in an easy,nontechnical way and builds it up again using his own unique methods. "Basically,the Clueless books are my ntes. Exactly the way I teach: friendly,clear. . . with some humore and plenty of emotion!"
Meant to bridge the gulf between the student,the textbook,and the teacher,Geometry for the Cluelessis packed with all the information you need to conquer geometry. This intensive study guide gives you:
"I am always delighted when a student tells me that he or she hated math. . . but taking a class with me has made math understandble. . . even enjoyable," says Bob. Now it's your turn. Sharpen your #2 pencils,and let Bob Miller show you how to never be clueless
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UCD School of Mathematical Sciences
Scoil na nEolaíochtaí
Matamaitice UCD
Access to Science and Engineering
This page is especially for students taking an access course with the UCD Adult Education Centre. This course prepares students for entry to university.
Welcome to the Access to Science and Engineering Mathematics webpage.
This webpage is an additional resource to access students, it is currently maintained by the tutor.
The mathematics component of the course aims to develop the basic mathematical skills necessary for science.
As the course progresses problem sheets will be posted here.
You are encouraged to seek out similar problems from text books or other sources.
For anyone who wants to see the expected standard of maths, feel free to view the sample exam papers provided.
I don't recommend it personally, but it reassures some people.
If it has been many years since you've been doing maths then exam papers might not encourage you!
The pace of the course should be gentle enough at first to allow everyone reach this level.
Here are links to some basic skills
worksheets that you may find useful - these are an additional resource and do not coincide with
course material. These contain very clear notes and exercises.
I recommend these as a supplement to the foundation maths text books.
In fact, for most purposes these may be better than a textbook.
Thanks to Dr Brien Nolan of DCU for his kind permission
to use them.
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For the modern student like you--Pat McKeague's INTERMEDIATE ALGEBRA, 9
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Description
FeaturesSix-Step Approach to Problem Solving - This tried and proven approach provides students with a systematic and logical framework for analyzing, comparing, estimating, and solving workplace applications.
Self-Study Exercises - These exercises conclude each section within a chapter, providing frequent opportunities to practice the concepts presented in the section.
Chapter Review of Key Concepts - These chapter-concluding summaries provide a quick overview of each learning outcome, the solution procedure, and an example - an excellent study tool prior to completing pratice sets or taking tests.
Additional Practice for Students - Additional practice materials, not found in the text, can be found on the text's companion website at
New Online Support:
MyMathLabwith MathXL
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MATH 093 5 credits Pre-Algebra The course is designed for students transitioning between arithmetic and algebra. Students will review arithmetic with real numbers, work with expressions containing variables, solve linear equations, graph linear equations in two dimensions, calculate slopes and intercepts for lines, and use unit analysis to solve applications. This course prepares students for Math 098. Prerequisites: "C" or better in MATH 090 or appropriate assessment score.
MATH 095 5 credits Thinking with Math Investigates the main ideas of algebra and geometry, with an emphasis on building math self-confidence and problem-solving ability through active experimentation and application. Prerequisite: MATH 090 or appropriate assessment score.
MATH 098 5 credits Elementary Algebra Topics include solving linear, quadratic (by factoring) and rational equations; solving a linear system of equations; manipulating polynomials (adding, subtracting, multiplying and dividing); and using exponent properties to simplify expressions. Students will also graph linear equations in two variables, calculate slopes, and find linear functions. Function vocabulary will be used. Prerequisites: MATH 093 with a grade of "C" or better or appropriate assessment score.
MATH 100A 5 credits Technical Math for Allied Health Applied mathematics course for allied health students. Students will learn the mathematics necessary for interpreting and computing dosages. Topics include fractions, percentages, measurement systems, unit conversions, oral, parenteral, IV and pediatric dosages. Topics from statistics may also be included. Not intended for students planning to transfer to a four-year college. Prerequisite: MATH 096 or MATH 098 with a "C" (2.0) or better or appropriate assessment score.
MATH 100T 5 credits Technical Math for Industrial Fields Applied course in mathematics for industrial fields. Topics include proportions, formulas, conversions, geometry and basic trigonometry and their applications to industry. Not intended for the student planning to transfer to a four-year college. Prerequisites: MATH 096 or MATH 093 with a "C" (2.0) or better or appropriate assessment score.
MATH 105 5 credits College Algebra Includes fundamental operations, factoring, linear and higher equations, functions and their graphs, inequalities, systems of equations, exponential and logarithmic functions and their relationship to the social and natural sciences. Prerequisite: MATH 097 or MATH 099 with a grade of "C" (2.0) or better or appropriate assessment score.
MATH 108 5 credits Mathematical Reasoning Emphasizes the mathematical reasoning process. Explores problem solving, sets and their properties, symbolic logic, and geometry. Additional topics can include counting techniques, probability, consumer math or other topics in the text. Course is for students seeking to broaden their appreciation of math. Prerequisite: MATH 097 with a grade of "C" (2.0) or better or appropriate assessment score. Evidence of competency in MATH 097 or MATH 099 is required for this course to transfer.
MATH& 141 5 credits Precalculus I Functions and their graphs (including elementary, exponential and logarithmic functions, and the conic sections) and their inverses in the context in which they are used in calculus. Work with graphing calculators will be integrated into the course. Prerequisites: MATH 105 with a grade of "C" (2.0) or better or appropriate assessment score.
MATH& 142 5 credits Precalculus II Introduction to trigonometric functions as they relate to the unit circle and right triangle. Graphs of the functions, applications, problem solving, identities, inverse functions, complex numbers, vectors and analytic geometry including polar coordinates and parametric equations. The basic concepts of sequences and series will be covered. Prerequisites: MATH& 141 with a grade of "C" (2.0) or better or appropriate assessment score.
MATH& 146 5 credits Introduction to Stats Fundamental concepts and applications of descriptive and inferential statistics. Includes measures of central tendency and variability, statistical graphs, probability, the normal distribution, hypothesis testing, confidence intervals, and regression analysis. Graphing calculator techniques are used throughout the course. Prerequisite: MATH 105 with a grade of "C" (2.0) or better or appropriate assessment score or instructor's signature.
MATH& 148 5 credits Business Calculus Differential and integral calculus designed for students majoring in business administration, social sciences and other programs requiring a short course in calculus. Work with graphing calculators will be integrated into the course. Prerequisites: MATH 105 with "C" (2.0) or better or appropriate assessment score.
MATH& 152 5 credits Calculus II Definite and indefinite integrals, techniques of integration. Application of the integral to areas, volumes and work problems. Derivatives and antiderivatives of the transcendental functions. Prerequisite: MATH& 151, with a "C" (2.0) or better.
MATH& 153 5 credits Calculus III More techniques and applications of integration. Parametric equations and polar coordinates, vectors and vector-valued functions, infinite series and sequences. Prerequisite: MATH& 152, with a grade of "C" (2.0) or better.
MATH& 171 5 credits Math for Elementary Educators 1 First of three math courses intended for elementary educators. Topics include number theory, mathematical problem solving, logic, real number systems, arithmetic operations and functions. Other topics related to math instruction at the K-8 level will be included. Hands-on activities are incorporated. Prerequisites: MATH 097 or MATH 099 with a grade of "C" (2.0) or better or appropriate assessment score. Evidence of competency in MATH 097 is required for this course to transfer.
MATH 200 5 credits Finite Mathematics Survey of the essential quantitative ideas and mathematical techniques used in decision making in a diversity of disciplines. Includes systems of equations and matrices, linear programming, finance, probability and its uses. Additional topics from Precalculus 1 may be included. Graphing calculators will be integrated into the course. Prerequisites: MATH 097 or MATH 099 with a grade of "C" (2.0) or better or appropriates assessment score.
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CKSD calculating new math requirements
The Klahowya Secondary School ninth-grader's feet can't quite reach the ground when he sits in one of the heavy, cushy chairs in the main office. When the red-haired freshman was in sixth grade, upon teacher recommendation he sped up his math studies and by seventh grade, he was in algebra 1.
"Normally I'd be in geometry at this age," said Kevin Hassett who is enrolled in the first all-freshman algebra 2 class at KSS.
Hassett says he's always been a fan of functions and sequences and sitting in Ellen Kraft's algebra 2 classroom has proven to be a complementary occupation to his advanced physical science class.
One reason he takes math is because he looks ahead to where the numbers could figure into his career.
"I think in my future jobs which might include engineer or something in the realm of science (math would be useful) and again I take it because I like it," Hassett said.
An appreciation of math like Hassett's adds up to perfection in Dave Thielk
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Algebra (M3.1)
LECTURER: V Murali
Description
Algebra is one of the main areas of mathematics with a rich history. Algebraic structures pervade all modern mathematics. This course introduces students to the algebraic structure of groups, rings and fields. Algebra is a required course for any further study in mathematics.
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introPrelude to Mathematics by W. W. Sawyer This lively, stimulating account of non-Euclidean geometry by a noted mathematician covers matrices, determinants, group theory, and many other related topics, with an emphasis on the subject's novel, striking aspects. 1955Product Description:
introductory text imparts other valuable mathematical tools and illustrates the intrinsic beauty and interest of mathematics. Journey into Mathematics offers a coherent story, with intriguing historical and etymological asides. The three-part treatment begins with the mechanics of writing proofs, including some very elementary mathematics--induction, binomial coefficients, and polygonal areas--that allow students to focus on the proofs without the distraction of absorbing unfamiliar ideas at the same time. Once they have acquired some geometric experience with the simpler classical notion of limit, they proceed to considerations of the area and circumference of circles. The text concludes with examinations of complex numbers and their application, via De Moivre's theorem, to real
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Reading This Could Help You Sleep: Caffeine in Your Body
Course Topic(s):
Developmental Math | Exponentials
"Reading This Could Help You Sleep: Caffeine in Your Body" is an introduction to exponential functions of the form (ab^t) at the Intermediate Algebra level, with emphasis on the meaning of these functions and their graphs. The concept of half-life is introduced. A conditional function (a piecewise-defined function) is used. "Get the Lead Out" extends the study of exponential functions and can be used to introduce the use of logarithms to "un-do" exponential expressions in solving equations. "So Much Coffee, So Little Time" can be used at the Intermediate Algebra level to help students see the value of "solving systems of linear equations". It shows connections between solving linear equations and exponential functions, and continues the study of exponential functions, this time of the form (ab^t + c). End behavior of these functions with horizontal asymptotes is included.
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Student Workbook for 'Mathematics Explained for Primary Teac
Trade review
Tried and tested by primary PGCE trainees, this workbook provides students with all they need to successfully review and use the material of the new edition of "Mathematics Explained For Primary Teachers".
Synopsis
Tried and tested by primary PGCE trainees, this workbook provides students with all they need to successfully review and use the material of Mathematics Explained for Primary Teacher, Fourth Edition
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The greatest compliation of math functions is back! Bigger, Better, more Powerful than ever! Just over 10k big (went down in size due to some major restructuring, but it does everything it did before but better, I promise), this program has 21 different functions including the quadrati, system of equations, point length/midpoint/slope, polygon angles, real number to hexadecimal converter, hexadecimal to real number converter, proportions, geometric mean, 30-60-90 triangle, 45-45-90 triangle, radical reducer, triangle type, and pythagorean theroem! and so much more! whew! so much for one program! A must have for any math student, smart or dumb! download it now before TiCalc's servers are overloaded!
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Lecture 25: Math 65 - Lesson 13a
Embed
Lecture Details :
Solving Quadratic Equations Using Square Roots
Course Description :
Introductory Algebra - Second Term Introduces algebraic concepts and processes with a focus on function, linear systems, polynomials, and quadratic equations. Applications, graphs, functions, formulas, and proper mathematical notation are emphasized throughout the course. A scientific calculator is required. The TI-30X II is recommended.
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For every student who has ever found the answer to a particular calculus equation elusive or a certain theorem impossible to remember, QuickStudy comes to the rescue! This 3-panel (6-page) comprehensive guide offers clear and concise examples, detailed explanations and colorful graphsall guaranteed to make calculus a breeze! Easy-to-use icons help students go right to the equations and problems they need to learn, and call out helpful tips to use and common pitfalls to avoid.
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Maths for Chemists
Synopsis
The two volumes of Maths for Chemists provide an excellent resource for all undergraduate chemistry students but are particularly focussed on the needs of students who may not have studied mathematics beyond GCSE level (or equivalent). The texts are introductory in nature and adopt a sympathetic approach for students who need support and understanding in working with the diverse mathematical tools required in a typical chemistry degree course. The early chapters of Maths for Chemists Volume I: Numbers, Functions and Calculus provide a succinct introduction to the important mathematical skills of algebraic manipulation, trigonometry, numbers, functions, units and the general grammar of maths. Later chapters build on these basic mathematical principles as a foundation for the development of differential and integral calculus. In spite of the introductory nature of this volume, some of the more important mathematical tools required in quantum chemistry are deliberately included, through a gradual introduction to, and development of, the concept of the eigenvalue problem. Ideal for the needs of undergraduate chemistry students, Tutorial Chemistry Texts is a major series consisting of short, single topic or modular texts concentrating on the fundamental areas of chemistry taught in undergraduate science courses. Each book provides a concise account of the basic principles underlying a given subject, embodying an independent-learning philosophy and including worked examples.
Reviews
A useful addition to the resources available for teaching mathematics to chemists. Source : "Education in Chemistry, January 2005 Issue (Paul Yates)"
"""... Undergraduates in biochemistry and all branches of chemistry, particularly students with a limited background in maths, will find this book essential. """ Source : " December 2003"
"""... The importance of mathematics in chemistry can not be under estimated; books aiming to show the many applications of the subject are always very welcome. """ Source : "Chemistry World, Vol 1, No 4, April 2004, p 59-60"
The mathematical ability of chemistry undergraduates continues to be an issue for many departments, so this new edition is a timely update to the resources available for both staff and students. Source : November 2012 | Education in Chemistry | 31
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This course provides an opportunity for students to see mathematics used in ways not emphasized in traditional algebra classes. The course emphasizes problem-solving. Specific topics covered will be selected from the following: set theory and logic, number theory and systems of numeration, unit conversions and dimensional analysis, consumer mathematics, probability, graph theory, and voting theory.
Prerequisite: Placement level of MA 113 or higher, or completion of MA112 with a C - or better. Students who have already received credit for MA 117 are not eligible to take MA 119.
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Purpose
This textbook and Internet resource provides introductory information, concept or skill development in Mathematics for grade 9, 10, 11, and 12 students who are at grade level in a single student situation.
Brief Description
Advanced Algebra II provides three complementary resources for teachers and students that combine to provide a friendly, easy-to-understand explanation of Algebra II concepts. The main text, "Activities and Homework", consists of a series of worksheets for both in-class group work as well as homework assignments. The concepts behind those activities are described in detail in the "Conceptual Explanations" text. The third book, the "Teacher's Guide", provides instructors with guides and suggestions for presenting these materials.
This content is available for free at the Connexions website ( For a guide on using Advanced Algebra II and a complete listing of topics, see
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Home Study Kit--Algebra 2, Third Edition
This Saxon Algebra 2 Home Study Kit includes the Student Textbook,
Testing Book and Answer Key. Traditional second-year algebra topics, as
well as a full semester of informal geometry, are included with both
real-world, abstract and interdisciplinary applications. Topics include
geometric functions like angles, perimeters, and proportional segments;
negative exponents; quadratic equations; metric conversions;
logarithms; and advanced factoring. Student Text is 558 pages, short
answers for problem/practice sets, an index and glossary are included;
hardcover.
The Test book contains both student tests and solutions with work shown along with the final answer. 32 tests are included.
The Answer key shows only the final solution for the practice and problem sets found in the student text. 44 pages, paperback.
Saxon Algebra 2 Kit & DIVE CD-Rom, Third Edition
Get everything you need for a successful and pain-free year of learning
math! This kit includes Saxon's 3rd Edition Algebra 2 textbook and
tests/worksheets book & answer key, as well as the DIVE Algebra 2
CD-ROM. A balanced, integrated mathematics program that has proven
itself a leader in the math teaching field, Algebra 2 covers geometric
functions like angles, perimeters, and proportional segments; negative
exponents; quadratic equations; metric conversions; logarithms; and
advanced factoring.
The DIVE software teaches Saxon lessons
step-by-step on a digital whiteboard, each lesson averaging about 10-15
minutes in length. Because each lesson is stored separately, you can
easily move about from lesson-to-lesson as well as maneuver within the
lesson you're watching.
The Saxon Teacher for Algebra 2, Third Edition on CD-ROM
Want some help with your Saxon Math? Saxon Teacher is designed
for you! Four discs feature lessons explained by a trained Saxon
Teacher utilizing a white board; students are walked through each
lesson, practice, & problem set. Problem Sets are accessed
individually, while Practice Sets are presented in one continuous video
so students can quickly review the teacher's solutions after working
the problems themselves. The fifth CD features step-by-step solutions
to each test question. Students view the lessons while following along
in the textbook, pausing to read certain sections on their own; example
problems are designed to be worked alongside the teacher. This kit
includes 5 CD-ROMs that will work on both Macs and PCs, but not TVs.
Bob Jones Geometry Grade 10 Homeschool Kit, 3rd Edition
BJU Press' Geometry text walks the student step-by-step through the
basic concepts of angles and degrees through more complex proofs and
theorems. Brief chapter introductions and photographs bring in an
element of real-world practicality, while biblically based material is
also to shed light on the purpose of geometry. Example exercises
provide additional help in recognizing theories, and instructions are
given for each new concept; exercises follow, and are given in sets,
with a cumulative review finishing off each unit. Concepts covered
include incidence geometry; subsets; segments & measurement; angles
and measurement; proofs; area; volume; symmetry; trigonometry and more.
Switched-On Schoolhouse 2012 Grade 10
Teaching Textbooks Algebra 2 Kit
Designed specifically for homeschoolers, this packet includes a spiral
bound 753 page teaching textbook, Answer Key & Test Bank Workbook,
set of 4 Lecture and Practice CD's, set of 6 Solutions CD's, and a test
solutions CD.
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This notebook illustrates how Mathematica is used in a 100-level college course in mathematical modeling, using the text Elementary Mathematical Modeling - A Dynamic Approach, by James Sandefur (Thompson / Brooks Cole).
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Maths Mutt
"Maths Mutt Mathematical Resources"
Maths Mutt provides Mathematical Resources for both teachers and students with notes , examples and quizzes covering courses from secondary 1 to Advanced Higher Mathematics with a breakdown of curriculum for excellence maths and numeracy experience and outcomes. Although the various topics are approached from the point of view of the curriculum in Scotland, content is suitable as a revision aid for anyone studying mathematics. There is also a handy bookstore and useful web links to further information and resources
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A self-paced, computer-based course covering the same material as MATH 085. Topics include introduction to variables and signed numbers, solutions to linear equations and inequalities, simplification of algebraic expressions, evaluation and manipulation of formulas, with an emphasis on word problems and graphics of linear equaltions. A scientific calculator is reuqired.
Prerequisite: MATH 075 with a minimu grade of C- or assessment at MATH 085 level and ENGL/ 085 with a minimum grade of C (may be taken concurrently) or equivalent. A student who has earned a grade of "U" in any CMATH course will not be allowed to take any other CMATH course.
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School of Engineering, University of Brighton, Cockcroft Building, Moulsecoomb, Brighton BN2 4GL, UK
Some new ideas in teaching mathematics to engineering students and the implementation of theseideas into the teaching of mechanical engineering students at Brighton University are discussed.The importance of explaining to the students why knowledge of mathematics is essential for their future practical work is emphasized. Mathematics is a language for expressing physical, chemical and engineering laws and general equations should be illustrated by practical numerical examples inorder to transfer the surface/atomistic approach to learning to the deep/holistic one. Necessarysteps in the manipulation of algebraic equations should be highlighted. Formal lecturers should besupplemented by compulsory reading, handouts, elements of small group teaching and formativeassessment. The analysis of self-assessment forms completed by students show that they learn physical concepts much easier than mathematical concepts.
INTRODUCTION
AN ENGINEERING student once said, `Mathe-matics is when numbers are put into equations'.This statement obviously contains an element of truth. One cannot expect engineering students toperceive mathematics in the same way as pro-fessional mathematicians usually do, yet the pro-fessional engineer must acquire not only empiricalbut also abstract understanding of mathematics. Itseems that the objective of teaching mathematicsto engineering students is to find the right balancebetween practical applications of mathematicalequations and in-depth understanding. In thispaper I discuss this balance and some practicalways of achieving it based on my experience of teaching thermofluids to engineering and energystudents at Brighton University. The achievedresults are discussed based on self-assessmentforms completed by the students.
MOTIVATION
It should not be taken for granted that engi-neering students understand the need to studymathematics in the first place. Although mysubject is not mathematics but thermofluids, itinevitably contains a number of mathematicalequations which I tried to explain in detail.When, after the first few lectures of the course, Iasked my students to complete feedback forms,about 80% of the students complained that mycourse was too academic. One of the students triedto describe this general mood by writing: We aremostly not academics but practical engineers; weforget what we are told but never forget what wesee or discover for ourselves!'It was clear that I made at least two mistakes indesigning my course. Firstly, the theory was indeednot properly balanced with practical applications.Secondly, the need for the theoretical part was notwell explained at the beginning of the course. I hadto put things right in order to complete the coursesuccessfully.There are obvious `natural' limits to the depth of the mathematical analysis. If we don't set theselimits we can, in theory, end up studying topology(the foundations of mathematics)
ad infinitum
. Ourbrain may be working very hard, but its directcontribution to the science of engineering would benegligible. On the other hand, there are not somany objects that can be physically touched inmodern engineering. For example, one cannot`touch' the boundary layer of a supersonic aircraftor the inside of a working internal combustionengine. In order to study them one needs todescribe them as abstract concepts in terms of mathematical equations.This means that mathematics is indispensablefor the engineering community, but the depth of itsstudy is bound to be limited. The best `practical'approach to mathematics is to understand it as alanguage for describing physical and chemicallaws. From this point of view understanding anengineering problem means the conversion of thisproblem into a physical and/or chemical problem,and its formulation in terms of mathematicalequations.Note that the fact that predictions of theoryagree with observations does not necessarilymean that the theory is correct. For example,Ptolemy's theory of the heavens was in goodnumerical agreement with observations over twomillennia. This, however, did not prevent it from
being wrong. (This idea was taken from themanuscript `What can we learn from numericalsimulations' by R. A. Treumann.) This means thata `practical' engineer cannot avoid the in-depthstudy of physics, chemistry and `practical' mathe-matics before applying them to engineering prob-lems. One cannot just take a mathematical modelas a `black box' and compare it with experiments.For example, a research engineer can find himself or herself severely hindered if he or she attempts toapply a computational fluid dynamics (CFD) codeto the solution of an engineering problem withoutunderstanding the underlying physical phenomenaand/or the limitations of the code [1]. I believe thatthis should be the main motivation for studyingmathematics for engineering students and it needsto be explained to students properly.Sometimes engineering students complain thatthey physically cannot perceive mathematical con-cepts. I believe that in this case the students can begiven the following formula:Result
Ability
Â
WorkEven the low ability students can almost alwayscompensate by hard work. This formula wassuggested to me by one of my own lecturers inmathematics. It can be generalized to:Result
Ability
x
a
Â
Work
x
w
where
x
w
b
x
a
or even
x
w
)
x
a
.What this is trying to say is that increasing theamount of work can easily compensate the limitedability of a particular student. Note that in manyreal-life situations lack of ability is confused withlack of confidence (see [2] for a more detaileddiscussion on confidence in learning). Obviouslyin the rare cases when the mathematical ability isclose to zero this cannot always be compensatedfor by hard work. Another factor which cancontribute to the result of learning mathematicsis the students' orientation to learning [3]. Forexample, students with personal or intrinsic aca-demic orientation, who enjoy exploring new andchallenging material are expected to get betterresults in mathematics than students with avocational or social orientation. The subject of mathematics requires higher levels of concentra-tion compared with other subjects, in general, andits immediate relevance to future students' jobprospects is not at first evident. Hence, studentswith vocational orientation do not have muchstimulus for this concentration and for them thefocus on practical elements of the mathematicalparts of the course is particularly important. It ismore difficult to accommodate students withsocial orientation in designing the course withoutsacrificing scientific and engineering standards.
NUMBERS AND FORMULAE
Ifengineeringstudentsareaskedtosolveasimpleproblem of finding the temperature distributionbetween two parallel plates at temperatures
T
1
and
T
2
provided that thermal conductivitybetween these plates is constant, some of themmight find it difficult. On the other hand, thesame problem can be reformulated in numbers:
Two parallel plates are kept at temperatures 200
8
Cand300
8
C6mapart.Thethermalconductivitybetweenthese plates is 10Wm
1
K
1
. Calculate the temperatureat the point which is 3m from the first plate.
In this case, almost everybody will promptlyanswer that the temperature is equal to 250
8
C.The reason for this is very simple. Most engineer-ing students think in terms of numbers rather thanin terms of abstract concepts. For this reason,students who experience difficulties with simpleanalytical calculations, can turn out to be verygood in practical applications.This does not mean that we should avoid dealingwith abstract concepts altogether for the reasonsalready discussed. This means, however, that everynew abstract concept needs to be accompanied byplentiful numerical examples.For example, if one just introduces Wien's lawfor blackbody radiation (
!
T
const) and moveson to the next topic, most students just forget itby the next lecture. On the other hand, if alecturer spends some time illustrating this lawby practical examples then it eventually registers.In other words, referring to referential aspects of students' experience, numbers lead the engineer-ing students from surface to deep knowledge (see[4, 5]). Referring to organisational aspects of theirexperience, numbers help make the transitionfrom an atomistic cognitive approach to a holisticapproach, that is, students start understandingthe problem as a whole, rather than concentrateon its parts [5, 6]. Note that the reverse processtakes place in the mathematical students com-munity: deep and holistic approaches are relatedin most cases to their concentration on formulae,while surface and atomistic approaches appear atthe stage of working with numbers (this obser-vation is based on my own experience as astudent, and the discussion of the matter withother students).
ALGEBRAIC EQUATIONS
When I started my course I assumed that thestudents felt comfortable with algebraic mani-pulations. After the first few lectures, however, Iunderstood that this assumption does not alwayshold. None of the students have problems insolving the equation
ax
b
and obtain the solu-tions
x
b
a
a
. However, if the same equation iswritten in a slightly more complicated way, say, as:
0
2
x
x
then its solution
x
a
0
2
causes diffi-culties among students if written straightaway.
S. S. Sazhin
146
Instead the left hand side of this equation needs tobe rearranged as:
0
2
x
x
0
2
x
x
where
0
2
, before its solution
x
a
iswritten. This normally takes just a few minutes,but if this is not done, then for many students thewhole lecture may be lost.Another problem with algebraic equations is thenotation. Whether we like it or not most studentstend to memorise equations in a particular nota-tion. Say, if the distance is indicated as
s
in onelecture, then this notation should be kept until theend of the course. I tried hard to persuade studentsto understand the structure of the equations ratherthan to memorize the notation (adopt deep ratherthan surface learning) but I had little success withmost of the group. My conclusion is that nota-tion needs to be unified to avoid any confusionespecially among the students who are at a novicelevel of skill acquisition [7].Finally, any sloppiness in the presentation of algebraic equations must be avoided by all means.Students do not easily recognize even the mostobvious printing mistakes and become stuck. Onmany occasions they tend to memorize and repro-duce wrong equations. The best solution to theproblem is to avoid sloppiness altogether. If amistake is found after the lecture it needs to beexplicitly admitted afterwards and not glossedover. The lecturer's handwriting is also veryimportant. One example in my experience iswhen one of my students copied the angle of attack of an aeroplane as 80 instead of 8
8
As aresult, he effectively dropped out of that particularlecture. I believe that the best way to tackle thehandwriting problem is to print formulae usingLatex software and show them to students usingtransparencies.
DIFFERENTIAL EQUATIONS
Differential equations, even relatively simpleones, seem to be a stumbling block for manystudents. My experience suggests that the simplestway of tackling this problem is to avoid it alto-gether by guessing the solution rather than solvingthe equation.For example, if we take the equation:d
2
y
a
d
x
2
0then it should be just proven that:
y
C
1
x
C
2
is its solution by direct substitution of this expres-sion for
y
into the original equation. Anotherapproach to the problem is to rewrite the originalequation as:d
z
a
d
x
0where
z
d
y
a
d
x
. Then it should be explained tothe students that the derivative of a constant iszero, so that the solution of the equation for
z
canbe written as:
z
C
1
Remembering the definition of
z
, this equationcan be rewritten as:d
y
a
d
x
C
1
Then students can be explained that derivativeof:
y
C
1
x
C
2
is equal to
C
1
. Hence
y
C
1
x
C
2
is the solutionof the original equation.Based on my experience, this rather lengthyapproach to the problem pays off and the studentsbegin to understand more complicated types of differential equations and their application toengineering problems. Note that on many occa-sions students are not confident in the concept of derivative itself. One teaching method of intro-ducing this concept is based on the discussion of velocity as a `natural derivative' [8, 9].
VECTORS AND TENSORS
Even simple manipulations with vectors, suchas summation and subtraction can cause prob-lems if students are not prepared for them. A one-dimensional problem could be a good startingpoint. One can consider the problem of calculatingthe velocity relative to the platform of two passen-gers in a moving train walking in opposite direc-tions to each other inside this train. This problemcan be easily visualised and students can recalltheir own experience. Students should reallybecome very confident with this simple problembefore they move on to the problem of summationand subtraction of vectors in three-dimensionalspace.The product of a vector and a scalar and thescalar product of vectors do not cause too manyproblems if they are explained in detail. The vectorproduct is often a stumbling point. In practice itseems to be more efficient to deal with the compo-nents of the vector product rather than with thegeneral equation written in vector form (cf. theGestalt theory as discussed by Laurillard [10]).The basic concepts of vector analysis such asgradient, divergence and curl can look ratherintimidating to some of the students. They can beintroduced if necessary, but it seems better to avoidany general manipulations. Instead, the analysiscan be focused on simple limiting (one or twodimensional) cases when the expressions for gra-dient, divergence or curl can be presented in simpleforms.Tensor analysis is normally excluded from theengineering curriculum altogether. This is regret-table since tensor is an essential and powerfulconcept for the analysis of many engineering
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Book Description: Elementary and Intermediate Algebra is a book for the student. The authors' goal is to help build students' confidence, their understanding and appreciation of math, and their basic skills by presenting an extremely user-friendly text that models a framework in which students can succeed. Unfortunately, students who place into developmental math courses often struggle with math anxiety due to bad experiences in past math courses. Developmental students often have never developed nor applied a study system in mathematics. To address these needs, the authors have framed three goals for Elementary and Intermediate Algebra: 1) reduce math anxiety, 2) teach for understanding, and 3) foster critical thinking and enthusiasm. The authors' writing style is extremely student friendly. They talk to students in their own language and walk them through the concepts, explaining not only how to do the math, but also why it works and where it comes from, rather than using the "monkey-see, monkey-do" approach that some books take.
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Mathematics is the only universal language that is spoken and used throughout the world. Mathematics provides all students with a powerful means of communication.
The mission of the Mathematics department is to give every graduate of the Stamford Public Schools the knowledge, understanding and skills they will need to compete in the 21st century world economy. Students will not only develop a comprehensive understanding of Mathematics but also will learn to effectively apply these concepts in a variety of problem solving strategies for the rest of their lives.
Important: The machine-generated language translation for this web page is provided for your convenience. It may contain grammatical or other errors and omissions.
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books.google.com.au - This book provides an account of those parts of contemporary set theory that are relevant to other areas of pure mathematics. Intended for advanced undergraduates and beginning graduate students, the text is written in an easy-going style, with a minimum of formalism. The book begins with a review of... joy of sets
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Over the past quarter century mounting evidence has pointed to the need for reform of U.S. undergraduate mathematics, especially the first two-year courses. For a century or so these courses have been largely the same at all U.S. colleges and universities, now numbering over 3000. With about three million enrollments each semester, the enterprise is large, complex, and firmly institutionalized in local, state, and national policies. Thus the stakes of any reform are high, but there are promising ideas for constructive change.
About three years ago I began implementing some of these ideas in our mathematics courses for future elementary teachers and that now is bearing fruit. At about the same time I began work on education for quantitative literacy (QL). Also called numeracy or quantitative reasoning, QL is the ability to identify, understand, and use quantitative arguments in everyday life and therefore a cultural field where language and number merge and are no longer one or the other. My focus here is the search for mathematics courses that are effective in education for QL. To better understand such a search and why it is necessary, it is helpful to look back a few centuries
Measuring Reality and Risk
About 750 years ago the idea of comprehending the world in quantitative ways—using numbers—was born (Crosby 1997). Mechanical clocks, marine charts, and double-entry bookkeeping—along with many other developments—for the first time provided ways to measure various aspects of civilization.
About 400 years after this beginning of quantification of Western society, in 1654, a French nobleman, Chevalier de Méré, who was fond of both gambling and mathematics, challenged the famed French mathematician Blaise Pascal to solve the two-hundred year old puzzle of how to divide the stakes of an unfinished game of chance between two players when one of them was ahead. Pascal turned to Pierre de Fermat, a lawyer and brilliant mathematician, for help, and the theory of probability ensued. For the first time people were able to make decisions and forecast the future with the help of numbers (Bernstein 1996).
In the 300 years following the introduction of probability, the management of risk and a multitude of other quantification systems became hallmarks of the new U.S. society. Historian Patricia Cohen wrote in the concluding pages of her 1982 book A Calculating People , " Numbers have immeasurably altered the character of American society. Our modern reliance on numbers and quantification was born in the scientific and commercial worlds of the seventeenth century, under the twin impacts of republican ideology and economic development" (Cohen 1982). In the two decades since Cohen's book, fueled by the development of computers, reliance on numbers and quantification has increased well beyond what could have been imagined, and no end is in sight.
In 1953, the Nobel Prize-winning economist and mathematician Kenneth J. Arrow described the complete market, a situation where every possible outcome of some scenario would be a commodity, for sale at a price (Surowiecki 2001). By 2004, much of Arrow's idea had been realized in the financial marketplace in what Mary Poovey calls the culture of finance where numbers and mathematics are used to reorganize the relationship between value and temporality (Poovey 2003). For example, risk, once time-dependent, is objectified, divided, and reassembled so that it can be traded. Stock options, derivatives, day trading, mark to marketing accounting, and adjustments to bad debt reserves are now among almost unlimited investment and accounting instruments available to individuals and companies. Understanding these very real almost everyday concepts is miles away for the student who struggles with meaning of odds or rates of change.
Mathematics and Measurement
Development of mathematics preceded the introduction of quantification – measurement – by more than 2000 years, but it was not until the fourteenth and fifteenth centuries that there was intermingling of mathematics and measurement. However, this intermingling was superficial, and over the subsequent centuries, until the present day, the real mathematics that mathematicians study, appreciate, and extend has remained essentially apart from commercial and other useful mathematics of the real world (Madison 2004).
Formal U.S. school mathematics from grades 8 or 9 through the first two years of college is dominated by a sequence of geometry, algebra, trigonometry, and calculus (GATC, for short). This sequence has been the essential offering in high school and college for over a century, with only minor changes in content coverage and grade-level offerings. During the middle part of the twentieth century courses for liberal arts students were constructed, mostly from selections from the GATC courses, and in the last twenty years data analysis and probability has been added to school mathematics. One of the courses originally conceived for liberal arts students, finite mathematics, now a standard offering for business students, is a mixture of probability and statistics, matrix algebra, sets, and logic.
Over the past century, while introductory college mathematics courses have changed little, major changes have occurred around them. First, U.S. society of the 21 st Century is vastly different from that of a century ago. Second, the college population now consists of the majority of typically eligible Americans while a century ago only a select few even finished secondary school. Third, remarkable technological developments have added potential cognitive power along with educational challenges about how to use the extra power. The quantitative demands on Americans for work, personal welfare, and citizenship have increased enormously. No longer is it acceptable to be mathematically or quantitatively illiterate, but there is convincing evidence that many, if not most, college graduates are unequipped for the quantitative demands they will face daily.
Quantitative Literacy
The QL that we should want for our graduates is what Lawrence Cremin referred to as liberating literacy—the power and habit of mind to search out quantitative information, critique it, reflect upon it, and apply it in their public, personal and professional lives (Cremin 1988). QL and mathematics are related, but they are not the same. Mathematics is an "abstract, deductive discipline, created by the Greeks, refined through the centuries, and employed in every corner of science, technology, and engineering" (Steen 2004). Although QL is not the same as mathematics or statistics, nonetheless, school and college mathematics and statistics, along with all other academic disciplines, bear responsibility for providing better education for QL. Part of that is developing better mathematics courses— ones that assist the learning of mathematics in real-world contexts and ones that help students develop the necessary habits of mind to handle the myriad of quantitative situations they will face. Better mathematics courses are necessary but not sufficient. Education for QL must be distributed across all disciplines in school and college in a coordinated way. Schools and colleges are responsible for both our students' quantitative education and the creation of the complexities of the society they face. Therefore, it behooves us to deal with our own handiwork.
The Search Begins
The mathematics courses required for The University of Arkansas Bachelor of Arts (B.A.) degree are College Algebra (MATH 1203) plus one of Finite Mathematics (MATH 2053), Survey of Calculus (MATH 2043), and Calculus I (MATH 2554). None of these courses was designed with
B.A. students in mind, and all four courses are dominated by mathematical methods and procedures. Very often students do not see the relevance of the content of these courses to their chosen major. Further, students are unlikely to develop the mental conceptual structures that appear necessary for long-term retention and use of the ideas and techniques of the course material. Since the courses focus on components of mathematics— e.g ., algebraic manipulation, matrix operations, derivatives and integrals—the courses are not strong in developing processes such as logical reasoning and problem solving, processes that are often more important than content knowledge in confronting unpredictable real-world situations. In spite of these apparent shortcomings, the situation is not very different in most U.S. colleges and universities. I believe we can do better, and last summer I was finally in position to try something that I believe will be better.
Over the past four years, I have worked with a national initiative to create some consensus on what constitutes QL in current U.S. society and how it can be achieved. In the process I collected many examples from U.S. newspapers and magazines of articles that require mathematics or statistics to understand and critique. I selected twenty or so of these articles and arranged them into eleven lessons entitled percent, petty thrift and buying stocks, condensed measures and indexes, lower math by Dave Barry, linear and exponential growth, measurement, visual representation of quantitative information, rates of change, weather maps and indexes, the "odds of that," and risk. Beginning in August, I conducted an experimental course (taught as a section of Finite Mathematics) based on these lessons. We are just finishing the first semester with twenty-six B.A. students, about half Journalism majors. I will use what I have learned this first semester and offer a second iteration of the course to 35-40 B.A. students in the spring of 2005, this time almost all Journalism students.
The course has several characteristics that distinguish it from many mathematics or statistics courses. These include:
Mathematics (including statistics, without saying it every time) is confronted, developed, and used as it occurs in the articles. The course is not organized by mathematical topics.
Almost all the problems are ill defined in the sense that assumptions are made that are not specified in the articles.
Estimation is often the most important lesson of problems.
Almost all exercises consist of gleaning information from the articles, formulating a mathematics problem, doing the mathematics, and reflecting the results back into the article. (The first, second, and fourth of these are difficult for students; doing the mathematics is easier for the students probably because they have practice at it. How to do the mathematics is also the easiest to teach because it is more structured and we have more practice at teaching that.)
Class sessions are casual and interactive. Students often work on group exercises. Every class begins with a discussion of quantitatively oriented newspaper or magazine articles that students have brought. My experience in this course indicates to me that education for QL requires that we change the way we teach mathematics as well as other subjects. The changes I see are rather substantial and include the following:
Mathematics should be encountered in many contexts such as political, economic, entertainment, health, historical, and scientific. Teachers will require broader knowledge of many of the contextual areas.
Pedagogy is changed from presenting abstract (finished) mathematics and then applying the mathematics to developing or calling up the mathematics after looking at contextual problems first.
Material is encountered as it is in the real world, unpredictably. Unless students have practice at dealing with quantitative material in this way they are unlikely to develop habits that allow them to understand and use the material. Productive disposition is critical for the students.
Considerably less mathematics content is covered thoroughly. Knowing arithmetic and being able to use it is better than knowing the techniques of calculus but having no ability to utilize them. Indeed, less can be more.
The mathematics used and learned is often elementary but the contexts are sophisticated.
Technology—at least graphing calculators with Computer Algebra System (CAS)—is used to explore, compute, and visualize.
QL topics must be encountered across the curriculum in a coordinated fashion. If I can coach writing then literature faculty can coach QL.
An interactive classroom is essential. Students must engage the material and practice retrieval in multiple contexts. This experimental course is only one of many steps we need to take to reform school and college education, especially general education. The complex society that we have created with the accomplishments of higher education has outstripped our facility at educating our students to live in this society. Our discipline-dominated higher education has served us well for a century but seems to need major restructuring to meet the challenges of educating all our students appropriately. The stakes are very high. As Carnevale and Desrochers (2003) point out, "The wall of ignorance between those who are mathematically and scientifically literate and those who are not can threaten democratic cultures.
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Algebra 1 Operations With Word ProblemBy:-Matthew David
Algebra is one of the most basic element of mathematics in which, we switch from basic arithmetic to variables. Here instead of using numbers we use different variables to represent different parameters. Algebra 1 is taught in initial level learning of algebra. They involve with basic terms like addition, subtraction and multiplication of variables, they also deal with finding the value of the variable. The algebra 1 operations with examples are illustrated in the following sections.
Algebra 2 Online Textbook And Generatorinatorics, and number theory, algebra is one of the main branches of pure mathematics
Algebra 2 Representation With Special FactorsBy:-Matthew David
A representation of algebra is a component meant to algebra. Now associative algebra is a sphere, If the algebra is not unital, then it might be prepared thus within a normal method here is no important differentiation among component for the resultant unital sphere, in that the individuality perform through the self plan, with illustration of the algebra.
Algebra PicturesBy:-Matthew David
Algebra is a branch of mathematics, which helps us to find an unknown quantity through mathematical operations. Algebra deals with unknown quantities, which can vary, that are called variables and fixed quantities that are constants. In Algebra, we use mathematical operators to group unknown quantities and form relations with unknown quantities, which are called algebraic expressions and equation. One of the grouping operations is raising the power of the variable or exponentiation.
we know 53=125. This means that logarithm of 125 to the base 5 is 3 and this is written as log5125 = 3
Algebra 1 Test Practice With ExercisesBy:-Matthew David
Algebra is defined as the part of mathematics which includes the study of laws of the operations that includes the equations and various structures including polynomials. In this article we are going to see some practice problems related to the algebra 1 test. Let us see some example problems for easy understanding of algebra 1 test practice.
Algebra The FunctionBy:-Matthew David
In algebra the numbers consider as constants, algebraic expression may include real number, complex number, and polynomials. In algebra several identities to find the x values by using this we can easily find the algebraic expression of the particular function. The sample algebra functions may include in the function of p(x), q(x),… to find the x value of the algebra the functions.
Fun Facts And Fun Games About AlgebraBy:-Matthew David
In math, algebra is a one of the most important part of the mathematics learning of the rules of the operations and relations facts. It includes about the polynomials, equations and algebraic structures. The algebra is combinations of about the analysis and number theory. Generally algebra is a one of the best way for learning about the elementary algebra facts. The elementary algebra is a basic form of the algebra.
Steps To Learn Basic AlgebraBy:-Matthew David
In universal algebra and mathematical logic, term algebra is a freely generated algebraic structure. For case, a signature consisting of a single binary relation, term algebra over a set X of variables is exactly the free magma generated by X. Term algebras play a role in semantics of abstract data types, where an abstract data type declaration provides the signature of a multi-sorted algebraic structure and the term algebra is concrete model of the abstract declaration.
Online Pre Algebra TextbooksBy:-Matthew David
Algebra is a branch of mathematics. Algebra plays an important role in our day to day life. Online pre algebra textbooks cover the four basic operations in algebra such as addition, subtraction, multiplication and division. The most important terms of algebra, variables, constant, coefficients, exponents, terms and expressions are explained in online pre algebra textbooks. We will know the symbols and alphabets in the place unknown values by online pre algebra textbooks. Hence, students can get online pre algebra textbooks.
Sample Pre Algebra Problems To DoBy:-Matthew David
Sample Pre-algebra is a common name for a course in middle school mathematics samples. Pre algebra problems deals with the Factorization of natural numbers, Properties of operations (associatively, distributives and so on), Simple (integer) roots and powers,Pre-algebra often the includes some basic subjects from geometry, mostly the kinds that further understanding of algebra and show how it is used, such as area, volume, and perimeter.
SI Unit Length And MeasurementBy:-Matthew David
Length implies the measure of how long an object or an path is thus we must possess some unit to measure such a important and major fact of our life because one cannot say that the object is 15 long or the path is 15 long because it does means that the object and the path have the same length but it might not true. Thus to measure length we have a specific unit which is called as the SI unit of length and it nothing but meter (m).
Resistance And Measurement SystemBy:-Matthew David
When we apply a potential difference between two ends of a conductor, an electric field will be set up inside the conductor and as a result current flows through the conductor.
The applied potential difference is proportional to the current flowing in the conductor. If V is the potential difference applied and I is the current
Scientific Method Steps And Acid Titration MethodBy:-Matthew David
A scientific method is comprised of asking and satisfying that question. This is made possible making by observations and performing experiments. The experiments executed experiment must be a "fair" test. A fair test refers to only if there is change in one variable, keeping all the other as constant.
Math Help GamesBy:-Matthew David
Generally in math help games, includes some of the play games like finding colors, missing the numbers. The math help games is used to calculate the numbers and finding some of the shapes like how many boxes, how many smiley's, how many balls and how many triangles, etc. Now we will play some of the games in math.
Word Problems In 2nd Grade MathBy:-Matthew David
Word Problems are the main chapter in 2nd grade math. These 2nd grades math involves basic arithmetic word problems. The basic arithmetic operations are add, subtract, multiply and divide. In this topic we have to seen about the addition and subtraction and multiplication and division word problems. Usually word problems are difficult here these 2nd grade math problems are very simple to study.
Math Square Root 12By:-Matthew David
In mathematics, we use a math square root symbol, which is known as radical. The design of radical symbol is (v). Rubicund is referred to a number which is present inside the root (i.e., here x is referred as rubicund). Square root design is deals with the different ways of expressing square root in various designs. Let us see what are the designs are there in square root in brief.
Solving Linear Quadratic SystemsBy:-Matthew David
Linear equation is an equation where the equation contains either constants or group of constant and a single variable. Linear equation has one or more variables.The degree of linear equation is 1.
Quadratic equation is also an equation where the highest degree is two. Quadratic equation general form is Ax2 + Bx + C = 0 where x = variable and a,b and c are constants.
Applying Word Problems In Linear FunctionsBy:-Matthew David
Let us see some of the word problems involving linear functions. In mathematics, the term linear function can refer to either of two different but related concepts: a first-degree polynomial function of one variable; in analytic geometry, the term linear function is sometimes used to mean a first-degree polynomial function of one variable. This function is known as "linear" because they are involving the functions whose graph in the Cartesian co-ordinate plane is a straight line.
One Step Linear EquationsBy:-Matthew David
In this page we are going to discuss about one step linear equations . A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables.(Source: From Wikipedia).
Algebra 1 And 2 Practice Problemsinatory, and number theory, algebra is one of the main branches of pure mathematics.
How To Perform AlgebraBy:-Matthew David
Algebra is the one of the important branches of mathematics that deals with the study of the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. In algebra we use letters or symbols to represent a number.Some of the classifications are Elementary algebra, Abstract algebra, Linear algebra, Universal algebra, Algebraic geometry, Algebraic combinatorics.
Algebra 1 Factoring SolverBy:-Matthew David
Algebra is one of the oldest and important topics of pure mathematics. Algebra was first stated by the Greek mathematicians. Algebra consists of constants and variables. Algebra is generally classified into algebra 1, algebra 2 and college algebra. Algebra includes linear algebra, non linear, polynomials, equations, radical expressions etc. When the concept of algebra is understood well then it is easy to solve the algebra problems.
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Mathematics
The Mathematics Department occupies the ground floor of the Academy Building and contains 8 classrooms each equipped with an overhead projector. Several rooms are equipped with interactive whiteboards. There is a Departmental Office which doubles as a store-room. The department also makes use of a small teaching room for the lowest teaching sets.
There are currently 12 full-time equivalent members of staff plus 3 members of the Leadership team who teach some Maths classes.
Course Outline/Overview
All pupils study Mathematics through Key Stage 3 up to GCSE and then around 20 take the subject at A-level in each of Year 12 and Year 13. At A Level, students have the choice of studying a Core Maths and Mechanics course or a Core and Statistics course. There is also the opportunity to take an extra A Level, called Further Mathematics, for appropriate students, as well as being prepped for Oxbridge entrance if required.
Key Stage 3
In Year 7 the focus is on teaching non-calculator skills and improving basic numeracy. Topics covered include the 4 rules of Addition, Subtraction, Multiplication, Division, Decimals and Money, Shapes, Graphs, Fractions and Percentages. As students progress through Key Stage 3 the content of their lessons will differ depending on which set they are in.
Throughout Key Stage 3, all students will sit past exam papers at regular intervals. Levels are used in their assessment throughout years 7 - 9 and targets will be based on these levels.
Mathematics is taught in sets from Day One in Year 7 based on Key Stage 2 results and CAT tests.
Key Stage 4
From 2006, mathematics at GCSE level will be taught in just 2 tiers, Higher and Foundation. Students doing the Higher tier can obtain grades from C up to A*. Students doing the Foundation tier can obtain grades from G up to C. Students in the lowest set will also take an Entry Level certificate as well as their GCSE. The examining board for this exam is Edexcel. Students will sit a final exam at the end of Year 11.
In general, the exam results for the department show good progress throughout every year group, leading to students overachieving when faced with external exams. The department and the school has a very supportive network in place so staff always have other colleagues to call upon in times of need.
Sixth Form
We follow the Edexcel A-level syllabus. All students take 3 modules in the Year 12 and 3 in the Year 13. Every AS Level student takes 2 Core modules, C1 and C2, plus their applied module, either Mechanics 1 or Statistics 1 as appropriate. In their A2 year, they will all take C3 and C4 plus their second applied module. We are also in our second year of offering Further Maths to A level. From our first group to complete the course, we produced the school's first Cambridge student, who has gone there to read Mathematics. Modules offered include all Mechanics, Statistics and Further Pure Modules, plus Decision 1, D2 and STEP preparation. Students do have the chance to resit earlier modules, but this is not encouraged. The cost of entry for any resit exam must be met by the student.
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Synopsis
An accessible treatment of the modeling and solution of integer programming problems, featuring modern applications and software
In order to fully comprehend the algorithms associated with integer programming, it is important to understand not only how algorithms work, but also why they work. Applied Integer Programming features a unique emphasis on this point, focusing on problem modeling and solution using commercial software. Taking an application-oriented approach, this book addresses the art and science of mathematical modeling related to the mixed integer programming (MIP) framework and discusses the algorithms and associated practices that enable those models to be solved most efficiently.
The book begins with coverage of successful applications, systematic modeling procedures, typical model types, transformation of non-MIP models, combinatorial optimization problem models, and automatic preprocessing to obtain a better formulation. Subsequent chapters present algebraic and geometric basic concepts of linear programming theory and network flows needed for understanding integer programming. Finally, the book concludes with classical and modern solution approaches as well as the key components for building an integrated software system capable of solving large-scale integer programming and combinatorial optimization problems.
Throughout the book, the authors demonstrate essential concepts through numerous examples and figures. Each new concept or algorithm is accompanied by a numerical example, and, where applicable, graphics are used to draw together diverse problems or approaches into a unified whole. In addition, features of solution approaches found in today's commercial software are identified throughout the book.
Thoroughly classroom-tested, Applied Integer Programming is an excellent book for integer programming courses at the upper-undergraduate and graduate levels. It also serves as a well-organized reference for professionals, software developers, and analysts who work in the fields of applied mathematics, computer science, operations research, management science, and engineering and use integer-programming techniques to model and solve real-world optimization
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Course Description
MATH 096 — Introductory Algebra (5 cr)
This course covers introductory algebra skills. Topics include signed numbers, linear equations, graphing linear equations, linear systems of equations, polynomials and rational expressions. This course is designed for students who need a review of high school algebra. Prerequisite: MATH 021 or 090 with a 3.0 or better within the last three years; or appropriate placement score. (SCC, SFCC)
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textbook represents an extensive and easily understood introduction to tensor analysis, which is to be construed here as the generic term for classical tensor analysis and tensor algebra, and which is a requirement in many physics applications and in engineering sciences. Tensors in symbolic notation and in Cartesian and curvilinear co-ordinates are introduced, amongst other things, as well as the algebra of second stage tensors. The book is primarily directed at students on various engineering study courses. It imparts the required algebraic aids and contains numerous exercises with answers, making it eminently suitable for self study.
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These
classes are special sections of Basic Math Skills, Elementary Algebra, and Intermediate
Algebra in which students will learn mathematics primarily using the computer
software rather than the traditional lecture/discussion method of instruction.
Tests will be done by hand in the usual manner and there may be (occasionally) short lecture/discussions
in classes.
The same materials are used in the computer-mediated and lecture sections (and distance learning sections) of the courses, except that some of the lecture teachers may not use the web software much and ALL of the computer-mediated classes and distance learning classes will use the web software. In the ACC bookstores, the shrink-wrapped package of materials for this course includes a folder with the access number for the web software called MyMathLab. If you purchase a shrink-wrapped package of materials, please don't open it until you talk with your instructor to be sure you are in the right course. Shrink-wrapped materials are NOT RETURNABLE after they are opened.
Used books will not contain access to MyMathLab and so students must purchase that separately for about $75. New books purchased in some way other than through the ACC bookstore may or may not include access to MyMathLab.
The computer-mediated course is taught in a computer lab with Internet
access and students use the computer software during the class. Students may
use the software outside of class in the Learning Labs at the campuses, and
students may use the software at home, provided the computer meets the minimum
requirements.
Students in these classes are in charge of their learning in a way that
is different from a traditional lecture class. The format of the course is somewhat
self-paced. This allows the student some freedom to set the speed at which he/she
works through the material, which means that he/she may be able to complete
the course before the end of the semester. It also means that students may spend
less time on topics with which they are already familiar and more time on topics
that are troublesome for them. Students will be provided with a weekly schedule
of topics to be covered and a schedule of exams. In order to complete the course
within the sixteen-week semester, students must generally keep up with the weekly
schedule and test schedule. In order to succeed in this class, students should
plan to spend about 9 to 12 hours each week working on the material, depending
on how much of the material is already review for them.
To determine whether a computer-mediated math
class is right for you, please complete the attached survey which is available
as either PDF
file or RTF file. Please be aware that
in order to view the PDF file, you must have Adobe Acrobat Reader. To download
it free of charge, please visit:
The RTF file should open with any text program, such as Microsoft Word.
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RELATED LINKS
Math Maneuvers
These Math Maneuvers take time and effort, but they work!
Read BEFORE class marking words, concepts, and examples that are not clear. Read with a pencil, paper, and calculator working out the examples in the reading. Check your progress on these with the solutions provided in the text.
Arrive to class EARLY with your text, pencil, notes, and calculator ready to go.
PARTICIPATE in class by actively taking notes, marking questions in lecture notes, and asking questions from the reading done before class.
Do homework problems AS SOON AS POSSIBLE after class. Some students like to RE-READ the text after class before beginning the homework. Get your lecture questions answered by the instructor (look for office hours on syllabus), a math specialist in the Learning Center, or a classmate.
CHECK ANSWERS to all homework, preferably every two or three problems. If you practice wrong, you may learn wrong, and you will need to unlearn the wrong way and learn the right way.
Find a STUDY PARTNER. When you are stuck, it may help to try to work with someone else. Even if you feel you are giving more help than you are getting, you will find that an excellent way to learn is by teaching others.
Work in the ACADEMIC RESOURCE CENTER in order to access the solution manuals, the math specialists, and other students.
Work during DROP-IN HOMEWORK HELP offered at special times in the Academic Resource Center.
KEEP UP with reading and homework DAILY. Math is sequential. Getting behind, even one day, tends to snowball downhill.
GETTING STUCK is to be expected. Don't get frustrated. Mark the problem, move on, and take the initiative to get help from a classmate, math specialist, or your instructor. Look at it as an opportunity to learn something new.
Cramming for math EXAMS seldom works. If you have kept up with homework, marking and getting your questions answered as you go, you should almost be ready for the exam. Set aside several hours to do a mixture of problems to be covered on the exam. Be sure to check answers to insure you are practicing correctly.
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Success in math includes mastery of geometry skills and requires children to make connections between the real world and geometry concepts in order to solve problems. Successful problem solvers will be ready for the challenges of mathematics as they advance to more complex topics. The activities in this workbook are designed...
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What exactly is the Golden Ratio? How was it discovered? Where is it found? These questions and more are thoroughly explained in this engaging tour of one of mathematics' most interesting phenomena. Veteran educators and prolific mathematics writers trace the appearance of the Golden Ratio throughout history and demonstrate a variety...
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Many of us trained mainly in the humanities and liberal arts may respect mathematics as an essential scientific discipline, but have done very little mathematics and often feel intimidated by its rigors. If you've ever wondered what mathematicians mean by the aesthetic elegance of their subject, here is your chance to...
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A Tour of the Calculus
Written by David Berlinski
Format: eBook
ISBN: 9780307789730
Our Price: $11.99Success in math requires children to make connections between the real world and math concepts in order to solve problems. Successful problem solvers will be ready for the challenges of mathematics as they advance to more complex topics. The activities in this workbook are designed to help your children catch up...
Read more >
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Perfect for either undergraduate mathematics or science history courses, this account presents a fresh and detailed reconstruction of the development of two mathematical fundamentals: numbers and infinity. One of the rare texts that offers a friendly and conversational tone, it avoids tedium and cont... read more
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A Source Book in Mathematics by David Eugene Smith The writings of Newton, Leibniz, Pascal, Riemann, Bernoulli, and others in a comprehensive selection of 125 treatises dating from the Renaissance to the late 19th century — most unavailable elsewhereElementary Number Theory: Second Edition by Underwood Dudley Written in a lively, engaging style by the author of popular mathematics books, this volume features nearly 1,000 imaginative exercises and problems. Some solutions included. 1978 edition.
History of the Theory of Numbers by Leonard Eugene Dickson Save 10% when you buy all 3 volumes of this set. Includes "Volume I: Divisibility and Primality," "Volume II: Diophantine Analysis," and "Volume III: Quadratic and Higher Forms."Product Description:
Perfect for either undergraduate mathematics or science history courses, this account presents a fresh and detailed reconstruction of the development of two mathematical fundamentals: numbers and infinity. One of the rare texts that offers a friendly and conversational tone, it avoids tedium and controversy while maintaining historical accuracy in defining its concepts' profound mathematical significance. The authors begin by discussing the representation of numbers, integers and types of numbers, and cubic equations. Additional topics include complex numbers, quaternions, and vectors; Greek notions of infinity; the 17th-century development of the calculus; the concept of functions; and transfinite numbers. The text concludes with an appendix on essay topics, a bibliography, and an index
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Mathematics
Overview
Welcome to the Munich International School Mathematics Department. Our department consists of 9 Mathematics teachers from around the world. We aim to deliver Mathematics instruction through lessons that are engaging and exciting for students. In addition to developing the skills necessary for students to be successful on their exams, and in post secondary Mathematics courses, we aim to enhance our students' learning through using technology to prepare them for a job market that is increasingly more technical.
We provide students with a wide variety of learning resources including but not limited to; interactive textbooks, notes, online course management systems (Moodle and Planbook), the internet and podcasts. One of our goals is to provide students with an appropriate level of challenge.
We participate in both the Canadian Mathematics Competition and the ISMTF Mathematics Competition.
We offer Mathematics Clinic which runs in one of the Mathematics rooms (S119, S120, S125 or S126). Maths Clinic is staffed by a member of the Mathematics department and is designed for students who feel they would benefit from extra assistance with their Mathematics. Students from all courses are welcome to attend and can bring questions from homework, classwork, quizzes and/or tests. Students may come anytime between 16:00 and 17:00 and stay for as long as they feel they need.
Grades 9 & 10
Mathematics Advanced
Mathematics Advanced is a rigorous programme intended to prepare students to take Higher Level Mathematics in the IBDP. In addition to covering the material in Standard Mathematics, supplementary topics of interest are explored. Students admitted to this program should have the recommendation of their grade 8 Mathematics teacher and achieve a 6 or 7 on the end of year report card with at least a 6 in Criteria A, and perform well on both the Canadian Mathematics competition and the end of year exam. Students must maintain a 'high' level of achievement to remain in the section.
Mathematics Standard
The Mathematics Standard programme is intended to prepare students for Standard Level Mathematics in the IB Diploma
Mathematics Foundation
With a low teacher-student ratio, students have the opportunity to explore basic Mathematics concepts in depth and engage in a variety of investigations to help them make the connection between Mathematics and the world around them. A major emphasis is placed on the use of technology and the Graphic Display Calculator. Admission to this program will be based upon the recommendation of the student's Mathematics teacher. The Mathematics Foundation programme prepares students for the Mathematics Studies program of the IB Diploma
Grades 11 & 12
IB Mathematics Higher Level
This is a demanding programme, requiring students to study a broad range of topics to varying degrees of depth. The programme caters to students with a solid background in Mathematics who are competent in a range of analytical and technical skills. The majority of the students enrolled will be expecting to include Mathematics as a major component of their university studies either as a subject in its own right, or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in Mathematics and enjoy meeting its challenges and engaging its problems. A major emphasis will be placed on the use of technology, and in particular, the use of the Graphic Display Calculator. The study of calculus makes up about ¼ of this syllabus.
Prerequisite: at least a 4 in Mathematics Advanced including at least a level 4 in Criterion A.
IB Mathematics Standard Level
A less rigorous course than IB Higher Level catering for students who anticipate a need for a sound Mathematical background in preparation for their future studies. The Mathematics SL course is a complete subset of the Mathematics Higher Level course. The majority of students likely to select this subject will be those who expect to go on to study subjects that have significant Mathematical content. In this course, in addition to a large amount of algebra, an emphasis will be placed on the use of technology and the Graphic Display Calculator for demonstrating and understanding most concepts. Students registering for this programme and wishing to attend a German university must enrol in a higher level Science. The study of calculus makes up about 25% of the course.
Prerequisite: at least a 5 in the Mathematics Standard including at least a level 5 in Criterion A.
IB Mathematics Studies Standard Level
This course caters to students with varied backgrounds and abilities in Mathematics. It is designed to build confidence and encourage an appreciation of Mathematics in students who do not anticipate a need for Mathematics in their future studies. Topics are designed to help students relate Mathematics to home, work, and leisure situations and as much as possible to other curriculum subjects. A major study of statistics is undertaken to assist students in becoming well-informed citizens. The majority of students selecting this subject are those whose main interests lie outside the field of Mathematics, and for many students, this will be their last formal Mathematics course. An emphasis on the use of technology and the Graphic Display Calculator will be made in this course.
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Numerical Methods in Electromagnetism
Electromagnetics is the foundation of our electric technology. It describes the fundamental principles upon which electricity is generated and used. This includes electric machines, high voltage transmission, telecommunication, radar, and recording and digital computing. This book will serve both as an introductory text for graduate students and as a reference book for professional engineers and researchers. This book leads the uninitiated into the realm of numerical methods for solving electromagnetic field problems by examples and illustrations. Detailed descriptions of advanced techniques are also included for the benefit of working engineers and research students.
Audience Students, engineers, scientists, and researchers involved in numerical methods for solving electromagnetic field problems in electrical machinery, as well as in high frequency devices; students and professors in electrical engineering departments.
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Microsoft Mathematics 4.0 for solving Mathematics problem
If you are a Windows user , you might have atleast come across the default calculator which is quite often used for some basic operations . But , here comes another program from Microsoft called "Microsoft Mathematics 4.0″ that helps the users to solve all kinds of maths problems . Whats more , its a free download and you can download Microsoft Mathematics 4.0 from the Microsoft Download Center .
Microsoft Mathematics provides a graphing calculator that plots in 2D and 3D, step-by-step equation solving, and useful tools to help students with math and science studies.
Students involved in any kind of Mathematical calculations might find this tool helpful . Students can learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and calculus.
Microsoft Mathematics includes a full-featured graphing calculator that's designed to work just like a handheld calculator. Additional math tools help you evaluate triangles, convert from one system of units to another, and solve systems of equations.
Another special feature of the Microsoft Mathematics 4.0 is that inclues a Ribbon UI . so the Microsoft Office users might find it really easy to use it .
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History of Modern Modern Mathematics Book Description
A survey of the major figures and mathematical movements of the 19th century, this is a thorough examination of every significant foundation stone of today's modern mathematics. Providing clear and concise articles on the fundamental definition of numbers through to quantics and infinite series, as well as exposition on the relationships between theorems, this volume, which was first published in 1896, cements itself as an essential reference work, a solid jumping-off point for all students of mathematics, and a fascinating glimpse at the once-cutting edge that now is taken for granted in an ever-changing scientific field. New York lawyer and mathematician DAVID EUGENE SMITH (1860-1944) authored a number of books while a professor of mathematics at Columbia University, including The Teaching of Elementary Mathematics (1900), A History of Japanese Mathematics (1914), and The Sumario Compendioso of Brother Juan Diez (1921).
Popular Searches
The book History of Modern Mathematics by David Eugene Smith
(author) is published or distributed by Cosimo Classics [1602063591, 9781602063594].
This particular edition was published on or around 2007-04-30 date.
History of Modern Mathematics has Paperback binding and this format has 84 number of pages of content for use.
This book by David Eugene
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9 Math
EQAO Grade 9 Math Test Preparation - a compilation of
hundreds of Algebra, Fractions and Math Word Questions exercises that target
what you need to know - from the Ask a Teacher web site (You may have to log in
either as a guest or create your own login before this link will work.)
Wired
Math - Gr. 9 - resources and online games for teachers,
parents, and students in grade 9 - from the University of Waterloo
Junior
High Math - This multimedia resource includes interactive math activities,
print activities, learning strategies, and videos that illustrate how math is
used in everyday life. The resource addresses all the intermediate mathematics
strands.
Math Continuum - a collection of interactive lessons from
Learn Alberta. (log in as Guest)
Webmath - This
site is composed of many math "fill-in-forms" into which you can type the math
problem you're working on. Linked to these forms is a powerful set of
math-solvers, that can instantly analyze your problem, and when possible,
provide you with a step-by-step solution, instantly!
Algebasics -
Clearly narrated examples
of understanding and solving algebra expressions make this an excellent site to
help introduce or reinforce basic to more complex algebra topics.
AskMe Algebra 1A and 1B coursesOnline tutorials from the University of Texas are a great way to
preview, practice, and review essential Algebra skills related to functions,
representing functions in multiple formats (words, tables, graphs, and symbols)
in the Four Corner Model, and using graphing calculators while practicing real
world math.
Javasketchpad Diagrams -
Diagrams drawn by the dynamic geometry software Sketchpad can be converted to
Javasketchpad diagrams. Teachers and students can work with these diagrams using
only a Java-capable browser. The diagrams on this page are specially designed
for Mathematics teaching and learning in secondary schools.
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Collaborators
Knovel Math for PTC® Mathcad®
Knovel Math, created in partnership with PTC and available as a Web-based service, provides fully documented and validated Mathcad worksheets of engineering calculations from trusted reference sources and delivers them directly into the Mathcad environment.
Generating mathematical equations is a common pain point among engineers, who often complain that they don't have the time or the comfort level to create their own calculations. Knovel Math includes worksheets for thousands of Mechanical, Civil & Structural Engineering topics.
The Science and Engineering of Materials is a leading Materials Science text book being taught in universities today. Knovel has collaborated with the Cengage and co-author Wendelin J Wright to create end of chapter problems which utilize searching on Knovel to reinforce and deepen students' understanding of the subject matter presented within the chapters.
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e-Learning at the YCDSB
MCT4C: Mathematics for College Technology, Grade 12, College Preparation
Grade 12, College
This course enables students to extend their knowledge of functions. Students will investigate and apply properties of polynomial, exponential, and trigonometric functions; continue to represent functions numerically, graphically, and algebraically; develop facility in simplifying expressions and solving equations; and solve problems that address applications of algebra, trigonometry, vectors, and geometry. Students will reason mathematically and communicate their thinking as they solve multi-step problems. This course prepares students for a variety of college technology programs.
MCT4C is based on Mathematics, The Ontario Curriculum, Grades 11 and 12, Ministry of Education, Revised 2007. It is comprised of 7 units, each of which has from five to 7 activities (lessons and assignments). The first 6 units are term work; the final unit is the culminating task
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Elementary emphasis on the practical applications of algebra motivates students and encourages them to see algebra as an important part of their daily lives. The student-friendly writing style uses short, clear sentences and easy-to-understand language, and the outstanding pedagogical program makes the material easy to follow and comprehend. The Fifth Edition places a stronger emphasis on problem solving, incorporating it as a theme throughout the text. 4-color hardback text w/complete text-specific instructor and student print/media supplement package AMATYC/NCTM Standards of Content and Pedagogy integrated in preview and Perspective Chapter Openers, Graded Exercise Sets, Real-World Applications, Writing Exercises, Group Activity/Challenge problems and Calculator Corners Boxes Many step-by-step worked-out examples provide students with detailed explanation throughout the sections Helpful Hints, Common Student Error Boxes, Chapter Summaries and Practice Tests included in each chapter help students study math and review for tests Cumulative Reviews are included in every other chapter and a unique Section 1.1Study Skills for Success in Mathematics, preps students with a study skills discussion and exercise set.
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From the Publisher "Remarks" boxes, definitions, and group projects. Written in a straightforward, readable, and helpful style, the book provides a thorough treatment of boundary-value problems and partial differential equations.
Description:
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.Book Format: Paperback. Number of Pages: 0323. Publisher: McGraw Hill Professional Publishing
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Book Description: This book exposes readers to many practical applications of geometry, especially those involving measurement. A three- part organization divides topics into Problem Solving, Geometric Shapes, and Measurement; Formal Synthetic Euclidean Geometry; and Alternate Approaches to Plane Geometry.
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Summary: This best-selling text balances solid mathematical coverage with a comprehensive overview of mathematical concepts as they relate to varied disciplines. The text provides an appreciation of mathematics, highlighting mathematical history, and applications of math to the arts and sciences. It is an ideal book for students who require a general overview of mathematics, especially those majoring in liberal arts, the social sciences, business, nursing and allied health fi...show moreelds. Let us introduce you to the practical, interesting, accessible, and powerful world of mathematics today-the world of A Survey of Mathematics with Applications, Expanded. ...show less
Hardcover Good 032150108X
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This essential resource provides support for Functional Maths, whether your students are taking the stand-alone qualification or GCSE Maths, or you simply want to bolster their KS3 maths programme of study.
Features
Produced in partnership with OCR and written by the senior examiners, so you can be confident it's spot on
Based on scenarios just like the real assessments, so your students can be fully prepared
Covers levels 1 and 2, also Entry 3, so catering for all your students' needs
CD-ROM format for flexible use, so you can
print out or use it on your school network/ VLE
Write-on worksheets so no laborious copying out
Full mark schemes and teacher guidance so you can help your students make real progress
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Mathematical Skills Center
The Math Skills Center offers two three-credit
courses for students who need to prepare for college level mathematics. The faculty and tutors of the Math Skills Center are dedicated to providing you the opportunities for success in studying mathematics. It is up to you to make good use of these opportunities. Our goal is to help you prepare for studying college level mathematics by mastering basic mathematical skills, developing effective study habits and achieving the confidence level needed for success. The operations of the Math Skills Center are based on your being an active student that takes responsibility for learning and is committed to achieving success in your college education. We look forward to working with you.
Starting Spring 2012 semester, the Math Skills Center (MSC) at SCSU implemented a policy for students repeating Math 070 or Math 072. This policy strengthens the MSC goal of helping students prepare for studying college level mathematics and will enhance student success. The MSC believes that students who participate in their education by attending class regularly and taking full advantage of support services by working at the Math Skills Center and attending Supplemental Instruction are most likely to achieve success. The following policy gives preference to students who demonstrate an active role in taking responsibility for his/her learning and a commitment to achieving success.
Students with a previous attempt at Math 070 or Math 072 will have a hold on future Math 070 or Math 072 registration. Our rationale is that incoming students and those with no prior attempts should be given priority in finding a seat in these classes.
Repeating students will be given priority for an override based on effort in the course including, but not limited to, attendance in the Math Skills Center, class attendance and participation in Supplemental Instruction. Those students not given priority may be required to successfully complete one semester of prep work before given permission to repeat.
Math 070 Jump Start is a course designed to prepare students for success in Basic Math Concepts, Math 070. Whether you are a student who struggles with the current pace of Math 070, are returning to math after a long interval, or would benefit from a refresher course using a non-traditional learning environment, Jump Start may be for you.
Specific areas of focus include basic conceptual and computational math skills in addition to organizational skills including time management, test taking preparation, and learning how to approach a math class and instructor expectations.
This course is free to students and self-paced with the guidance of an Undergraduate Learning Assistant. Jump Start is not used to calculate GPA, nor will you receive a grade, and it is not a replacement for Math 070, rather it helps students prepare for Math 070.
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Quick Links
Resources
Math Department
General Math Classes
ALGEBRA I
The purpose of this course is to provide students with the foundation for more advanced mathematics courses and to develop the skills needed to solve mathematical problems. Course content includes sets, variables, real number systems, equations and inequalities, relations and foundations, graphs, rational and irrational numbers, and radicals.
ALGEBRA II
The purpose of this course is to providestudents with a foundation for applying advanced skills to other mathematical and scientific fields. Course content includes linear and quadratic equations, factoring of polynomials, graphing systems of equations, and rational/irrational functions.
GEOMETRY
This course emphasizes critical thinking involving the discovery of relationships and their proofs, and skills in applying the deductive method to mathematical situations. Course content includes logic and reasoning, the study of Euclidean geometry of lines, planes, angles, triangles, similarity and congruence.
ANALYTIC GEOMETRY
Students will use graphing technology to find approximate solutions for polynomial equations. Write the equations of conic section s in standard form and general form, in order to identify the conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.)
TRIGONOMETRY
The purpose of this course is to teach students to make connections between right triangle ratios, trigonometric functions, and circular functions.
MATH FOR COLLEGE READINESS
The purpose of this course is to strengthen the skill level of high school seniors who have completed Algebra I, II, and Geometry and who wish to pursue credit generating mathematics courses at college level. Course content includes: Functions and Relations: Polynomials; Rational Expressions and Equations; Radical Expressions and Equations; Quadratic equations and Strategies for College Readiness. Exit requirements must pass with a "C" or better. Pass CPT: above 86, SAT: above 520, ACT: above 22. If the students meets the above exit requirements, the student will not have to take any prerequisite classes and must enroll in College Algebra within two years of completion of this course.
PRE-CALCULUS
The purpose of this course is to enable students to develop concepts and skills in advanced algebra, analytic geometry, and trigonometry. The content should include: trigonometric functions and their inverses, trigonometric identities and equations, vectors and parametric equations, structure and properties of the complex number system, polar coordinate system, sequences and series, concept of limits, conic sections, polynomial, rational, exponential, and logarithmic functions, and matrix algebra.
College Level Math Classes
AP CALCULUS AB
AP Calculus AB is designed to familiarize the student with the basic concepts of introductory calculus. It is equivalent to a first semester course in college. Course content includes functions, limits and continuity, derivatives and integrals. There is a strong emphasis on application through topics such as related rates, optimization, volume of rotation work and moments.
AP STATISTICS
The purpose of this course in statistics is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: Explore Data: Describing patterns and departures from patterns, Sampling and Experimentation: Planning and conducting a study, Anticipating Patterns: Exploring random phenomena using probability and simulation Statistical Inference: Estimating population parameters and testing hypotheses.
Math Electives
INTENSIVE MATHEMATICS
The purpose of this course is to enable students to develop mathematics skills and concepts through remedial instruction and practice. The content includes, mathematics skills that has been identified by screening and individual diagnosis of each student's need for remedial instruction and specified in his/her Progress Monitoring Plan, critical thinking, problem solving, and test-taking skills and strategies. This course counts as an elective.
MATH FOR COLLEGE SUCCESS
The purpose of this course is to prepare students for entry level College Mathematics. Major topics include properties of integers and rational numbers, integer exponents, simple linear equations and inequalities, operations on polynomials including beginning techniques of factoring, introduction to graphing, and introduction to operations on rational expressions. The content should include: using signed numbers; simplifying algebraic expressions; solving algebraic equations; simplifying exponents and polynomials; factoring polynomials; graphing linear equations; simplifying, multiplying, and dividing rational expressions; simplifying and performing operations with radicals. Exit requirements: Obtain grade of "C" or better and pass the state exit exam at the identified "cut-score". If the student meets the above exit requirements, the student will not have to retake the CPT or have to take any non-college credit classes within two years of completion of this course.
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Transition Mathematics
Main goal: The main goal of Transition Mathematics is to act as a stepping-stone between the processes learned in Pre-Transition Mathematics or Everyday Mathematics 6 to the material presented in UCSMP Algebra and UCSMP Geometry. Transition Mathematics incorporates applied arithmetic, algebra, and geometry; and connects all these areas to measurement, probability, and statistics.
Main theme I: Arithmetic skills and concepts are reinforced by continuous instruction in the uses of the four basic operations of addition, subtraction, multiplication, and division. Basic skills and number sense practice are reinforced by applications and the conversions among decimals, fractions, and percents, with both positive and negative numbers.
Picturing multiplication by 2.5 and by 0.8
Main theme II: The algebra in Transition Mathematics begins with the uses of variables in formulas, as pattern-generalizers, and as unknowns in solving problems. Graphing lines in the coordinate plane and the solving of linear equations and inequalities are developed.
Main theme III: The geometry in Transition Mathematics includes the use of transformations to demonstrate congruence, similarity, symmetry, and tessellations. Length, perimeter, area, and volume are studied as general concepts and with specific attention to common two- and three-dimensional figures. Drawing and constructions with and without the use of technology are both strongly encouraged throughout the text.
Comparisons between this and earlier editions: The reality orientation of the material and the overall approach of this groundbreaking book remain. Some of the content in the first and second editions of Transition Mathematics has been moved to Pre-Transition Mathematics due to (1) the existence of Everyday Mathematics and the general increase in the performance of students coming into middle school, (2) increased expectations for the performance of all students in both middle and high schools and the concomitant increased levels of testing, and (3) recommendations for more algebra and geometry in middle school courses preceding year-long courses in algebra and geometry. Calculators with graphing and list features are introduced early as pattern-fitting and problem-solving tools. Spreadsheets and dynamic geometry systems are found in activities throughout the materials. Students are engaged and learning is reinforced with the use of games.
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Algebra 1
Instructor's Annotated Edition
ISBN/ISBN10
978-0-495-38988-0
0-495-38988-9
Order#
1090340
Price
$202.75
Quantity
This special version of the complete student text contains a Resource Integration Guide to using the ancillary teaching and learning resources with each chapter of the text, as well as answers printed next to all respective exercises. Graphs, tables, and other answers appear in a special answer section in the back of the text.
Titles marked with asterisk (*) indicate product is restricted from sale to individuals and may only be purchased by a registered institution. Go here if you are not already logged in or need to register.
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Basic Math for Computer science engineer - Books/Videos - MathOverflow [closed]most recent 30 from Math for Computer science engineer - Books/VideosAMBROSE2010-07-16T14:45:51Z2010-07-16T14:45:51Z
<p>Hi All,</p>
<p>I am a Computer scienece engineering graduate working as a Technical Lead in a Software firm .</p>
<p>My day today work deals development of application which is always has limited time.</p>
<p>Now after some years almost don't remember the academics like basic math required for Computer graduate .</p>
<p>I want to develop my own product so wanted to think in terms of algorithm and maths .Before that wanted to refresh or re-learn the academics stuff .</p>
<p>Can anyone suggest the required math books ? (Remember I have already quoted "almost don't remember the academics" ).</p>
<p>If some could understand what I am asking for ., please provide the requested .</p>
<p>Thanks in Advance ,
Ambrose J</p>
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QuickMath Automatic problem-solving site that
let students enter an expression and get an answer to math problems dealing with
polynomial factoring, multiplication, long division, integration and differentiation.
S.O.S. Mathematics Site acts as a hyperlinked
math textbook. Students who have trouble following either the textbook explanation
or their teacher's lecture can visit this site for an additional explanation of
a difficult concept.
Hotmath This site offers free tutorial solutions
to the odd-numbered homework problems from most popular math textbooks. The tutorial
solutions seek to mimic what a tutor or teacher would say if a student asked for
help on the problem.
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Purcell has tougher problems that really reinforce the concepts but keep in mind that it is a first year text. It wont cover more advanced things, in particular the entire second half of Griffiths text. You need Griffiths for the more advanced material.
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Find a RillitoAll these experiences have enriched my understanding of math and my ability to explain math to my students. I want students to rise to the highest level of which they are capable. Learning math is more than just learning formulas or repeating a procedureWhile my software of choice for Desktop Publishing is Adobe InDesign, sometimes cost or compatibility becomes an issue. In these cases, Microsoft Publisher is a powerful tool to which many people, especially in an office running Microsoft Office, already have access. I have used Microsoft Publi...
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Intermediate Algebra
This Intermediate Algebra textbook is designed as a chronological course to guide you through High School Algebra (sometimes called Algebra II in some locations). This textbook assumes you have completed Arithmetic and Algebra. Although not required, Intermediate Algebra is normally taken the year after Geometry.
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ALEKS is an acronym for "Assessment and LEarning in Knowledge Spaces." This
Internet-based mathematics program can be used as a tool for Higher Education,
as K-12 targeted instruction correlated to state standards, or as an independent
use curriculum. For the purpose of this review, it has been used as an independent
use curriculum--as a personal tutor, which is the suggested use for homeschoolers.
Users have access to a full course library without a textbook requirement. Paper
and pencil and a variety of other mathematical tools may be useful to the student
when working certain problems, but answers, grading, and actual coursework are
done on the computer. Internet access is required to use `, but this
also enables students to use the program at different locations when convenient.
Courses are available for students in grades 3-12 and beyond, and available course
offerings range from elementary-level titles to middle school Geometry and Pre-Algebra
as well as high school Algebra, Geometry, Algebra 2, Pre-Calculus, and Trigonometry,
just to name a few.
Two or more accounts are set up for use of the program: a master account for use by the parent and an account for each student user. Each account has its own login name and password. The master account includes quizzing ability and automated reports so that parents can monitor progress. It also features a communication system that allows parents and students to leave reminders or other information for one another. The purchaser pays to use the program on a month-to-month, six-month, or annual basis. If a student completes the course requirements before the purchased time is entirely used up, parents can simply move the child into another course of their choosing. A User's Guide is available in both Web versions and PDF for parents to peruse at their leisure, and customer service is readily available. A Training Center is also available, with instructions on each aspect of using ALEKS.
ALEKS targets gaps in student knowledge determined from an initial assessment and adjusted throughout the use of the program. Upon first logging in, the student is led through a tutorial that explains the use of the program. Then the student begins the initial assessment, usually consisting of 20 to 30 unique questions determined by the previous answers of the student. After the initial assessment is completed, the student is presented with a pie chart that displays material already mastered, material that still needs to be mastered, and areas the student is actively learning. This visual presentation of what the student knows can be incredibly motivating throughout the course. From this point on, the student is able to select various topics to work on, based on the information in the pie chart. He or she will continue to see improvement on the chart as mastery is achieved. ALEKS continuously updates the chart and systematically adjusts questions and content to target the student's specific needs.
Parents monitor progress via reports emailed periodically from ALEKS. Also, by logging into the master account, parents can monitor attendance, topics worked, and the student's pie chart quickly and easily. The program even provides a projected time to completion, based on the student's rate of work. This information is incredibly helpful when determining exactly where your child is in the process of mastering the material. A feature called ALEKS Quick Tables may also be assigned by the parents, with a particular focus on only addition, subtraction, division, or multiplication facts. Parents can design and print individualized worksheets with specific details as to the number and types of problems. Visual tracking of progress is also provided in Quick Tables; a color-keyed table displays exactly what the student currently knows.
Our family finds ALEKS a wonderful addition to our homeschool. In our busy household we have children ranging from toddlers to those about to graduate from high school. So having a program that provides math mastery for independent learners is a tremendous help. Our daughter had easily cleared our local Algebra exit exam, but I knew she did not know the material to the level we desired for her. An adequate grade is not what we wanted; we wanted her to have mastery of the subject. ALEKS provided for this need.
A child without adequate independent study habits might need closer monitoring when using ALEKS, but the program certainly allows for that. Parents could assign a certain amount of time to be spent daily and then monitor whether the child was fulfilling that requirement. If your student simply hates work on the computer, ALEKS may not be the best choice as a math program. Likewise, those who require the use of math manipulatves will not find ALEKS an ideal learning tool. Also be aware that a child needs to be reading at least at a third-grade level before attempting this program.
Math skills are essential in all walks of life, and the assurance that our students are achieving mastery is truly priceless to our family. The flexibility of either a monthly, bi-annual, or annual commitment makes ALEKS a workable option for most homeschools. The actual price is comparable to other full courses offering the same information. Our family really needed a program that would allow me to focus on younger children while the older children studied independently. ALEKS filled that need beautifully. I encourage you to try ALEKS through their trial offer to see if it will work for your family. You will certainly enjoy knowing that your students are mastering math with greater understanding than any pile of worksheets or five-pound textbook could offer.
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The asymptotic approximations are not valid for small samples. The ability to perform exact statistical tests frees one from the worries of the quality of asymptotic P-values. `Exact Test` always gives you exact P-values, regardless of your...
Are multiplication tables showing up in your kid`s homeworks?
MULTIPLEjm will help your kids learn multiplication tables. Trying to beat their high score, they will have fun and will learn their multiplication tables.
Dicom is the first ever calculator of its kind that deals with dimensions of physical quantities in mathematical expression. Dicom enables calculations, conversions, and documentations in one place, at absolutely no cost.
Free math/graphing program that will allow you to develop and visually analyse mathematical expressions quickly and easily. Boasts a unique graphing module that permits expressions with up to 8 parameters to be plotted directly, such as y=Asin(kx+b).
Robot4 (tm) is a Robotic Arm Movement program where the arm is moved from given position to desired position(an Inverse Problem). The program finds the angles necessary for the desired position. Improved Productivity demo do to Calculus programmingMiddle-School (grades 5 through 9) math program written to provide skills in context. Students use coordinate graphing to guide a moving dot through a series of mazes. Think: coordinate graphing pinball.
Middle-School (grades 5 through 9) math program written to provide skills in context. Students calculate the probability of three different types of outcomes on a slot machine: three apples, at least one apple, or no apples.
Middle-School (grades 5 through 9) math program written to provide skills in context. Students write and solve simple algebra problems, then manipulate the vertices of an on-screen triangle so that it matches given information about its angles.
Middle-School (grades 5 through 9) math program written to provide skills in context. Given the radius or diameter of a circle, and the measure of a central angle, students find various areas and lenths correctly
CFB Primes is an optimised PalmOS program which can find the prime factors and closest primes of any integer up to one billion (US) in a few seconds or less. The standard edit menu allows you to make use of the results in other applications.
The Number Base converter is a tool which converts numbers from one base to another such as binary,decimal,hex,octal. The program is quite simple so even if you are a beginner you can use it without any trouble.
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expression can be modified or corrected, and recomputed at will. There ara also unit conversions, physical constants, computer math (binary, octal, hex), and function graphs included. For more advanced users, Kalkulator provides less trivial operations: numerical integration and differentiation, interpolaton (linear or cubic spline), statistics (mean, s.d., histograms, polynomial regression), column operations on stat data, polynomial roots, linear algebra (vector/matrix operations and systems of linear equations). Hard-to-find features: systems of non-linear and differential (ODE) equations, multi-argument function extremum search and point or histogram fitting with an arbitrary function. Most importantly, advanced features don't obstruct the basic ones. You can use as much of Kalkulator's capabilities as you need, without being hampered by the program capabilities you do not need. Disk save/restore, Help file and other niceties are also included. Kalkulator has been evolving on various OS platforms for the last 15 years, being refined and polished through all this time, and its author has more than 25 years of active experience in numerical methods and software development.
JMNumerics - JMNumerics is a Numerical Library for .NET. The library is written in pure C#, it has more than 150,000 lines of code with the most advanced algorithms for Linear Algebra, Differential Equations and Optimization problems.JMNumerics is a Numerical...
GNU OctOctNature of Equations Video - A video explaining the nature of equations in algebra.A video explaining the nature of equations in algebra.Midget - Tiny calculator with 12 variables, 32 functions, unit conversion, deg/min/sec...A tiny, but very capable numerical calculator for Windows. It differs from others in the way you calculate your expressions: enter the whole expression at a time, like...
Mr. Matt - An addictive logic puzzle game for Windows, all agesAn addictive logical puzzle game for Windows, similar to the classic Boulderdash, but with many enhancements and extra touches. Teen to adult age bracket, no gore or violence. You are Mr. Matt,...
Statmaster - Many scientific and engineering programs generate numeric output which needs to be presented graphically.Many scientific and engineering programs generate numeric output which needs to be presented graphically. Statmaster is a simple way to do...
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The materials below were developed for both graduate and undergraduate courses taught under the title "Mathematical Modeling." The philosphy of these courses is to cover as broad a range of topics with a modeling flavor as possible. It it generally easier to get a deeper knowledge of a subject X where one has seen the key ideas and some of the major results than it is to start off reading in a book devoted only to the content of topic X. Both issues involving modeling and situations where mathematics can be used to get insight will be considered. Some of these materials have been developed with the assistance of Stuart Weinberg, Teachers College.
Some of these items were also developed in conjunction with the P-credit courses offtered by Mathematics for America in conjunction with its Professional Development and Outreach in the New York City area.
Here are links to two very nice expository articles (pdf files) about how Gale/Shapley is used in the real world market of pairing medical students to hospitals where they can carry out there residency. The articles appeared original in SIAM News in 2003.
Here are two sets of "mechanical" exercises to cement your skill in solving bankruptcy problems and in using a variety of different methods to decide the winner of an election which involve preference ballots.
This is a brief list (with apologies to all the other wonderful books in this area) of important books about games, fairness, and elections. It also includes the important desirable features of a fair division scheme.
Mathematical modeling has interesting connections to other topics mathematicians and mathematics educators are interested in, namely, problem solving and estimation. In honor of July 4th here is a note with a poll and activity to probe the issues related to modeling, estimation, and problem solving.
There are many measures of central tendency in statistics (which single number "presents" a data set "best." Examples include the arithmetic mean, geometric mean, harmonic mean, mode, median, and mid-range value. When one wants to locate a facility (medical center, public lavoratory, fire house, etc.) one often desires to choose a "central" location so perhaps it may not be surprising that there are connections between statistics and facility location issues.
It may be worth the time to practice your understanding of the plurality, run-off, sequential run-off, Borda count, and Condorcet methods by doing the problems here. There are also questions that get at whether there might be "general theorems" involved in questions related to elections.
Elections and voting are the cornerstones of a democratic society. There are an amazingly large number of settings where American are involved with voting. We vote for local, state and federal officials, in some cases judges, and within our workplaces and educational institutions. There are also votes within legistlatures, clubs, and other "group structures." One can spend a whole lifetime studying mathematical insights into voting and elections. The mathematics involved covers the full gamut of mathematical tools.
In order to makes decisions one has to consider the various choices of actions that one can take and be able to understand which of these actions you consider as better. There are many aspects to this choice environment and this activity about probing your personal feelings about particular kinds of fruit my help see the complexities involved.
Here is a one one page introduction to the idea of a graph model. These are geometric diagrams which consist of dots (representing objects) and line segments which indicate relationships between the objects.
You can download a pdf file version of five modeling questions (each one of which can typically be solved in a single high school or college classroom period. The problems are based on the same "data" which consists of a grid graph with 6 sites singled out. The problems give rise (when scaled to realistic size, and to other settings) to heavily studied questions in "urban" operations research. You can find html versions of similar things if you scroll down to the urban operations research section below.
This activity is to suggest one of the many urban operations research problems which involve the routing of vehicles. This example centers around that gas must be delivered from a "depot" to the individual gasoline stations that get their gas supplies from this depot. This activity asks some questions but does not actually discuss the algorithms that have been developed to solve these kinds of problems. This activity is related to issues about bin packing and the traveling salesman problem.
This brief note discusses some ideas related Modeling Situation 1 and the modeling process in general. Specifically, the issue of finding data and information that is used in model construction is raised.
This brief note discusses some ideas related Modeling Situation 2. Different election decision methods give rise to different winners and this means that one has to try to think of ways to assess the pros and cons of different methods. Many nice lessons for K-12 education can be based on election methods and voting ideas. When there is no Condorcet winner for an election what should be done to make the election method "decisive." When there is a Condorcet winner is this the best method? There is also the chance to practice the construction of the pairwise preference matrix and the anti-plurality method is mentioned.
In many social choice and game theory settings there are payoffs for the outcomes to the participants. While these payoffs sometimes can be thought of as money, psychologically the same amount of money can mean different things to different people. The concept of tility is an attempt to deal with these complexities.
This note talks briefly about Arrow's Theorem, fairness axioms for election decision methods, and strategic voting. Strategic voting refers to voting for a ballot which reflects something other than your sincere feelings because it will give you a more favored outcome. This can be done when you know what the decision method being used is, and when data about how other voters might vote is available.
This note talks about an additional fairness condition, called Majority, that some would say should be obeyed by a "reasonable" election decision method. Although much more could be said about "election decision methods" we will move on to another phenomenon that occurs in voting situations. Namely, that votes are being taken by "players" (legislators) who represent groups of different population or economic power. To deal with situations of this kind, weighted voting is sometimes used. The idea is to have each player cast a "block" of votes at once, called the weight of the player. This situation arises in the Electoral College and many of the governing bodies of the European Union.
Here you can get a chance to practice problems involving the concepts of winning and minimal winning coalitons in a weighted voting game, as well as computing the Banzhaf and Shapley voting power for a weighted voting game.
This note has all the inequivalent wieghted voting games (which make some sense in practice) with 4 or fewer players. The weighted voting, minimal winning coalitons, and Shapely-Shubik power vector for each game is given.
This note proposes an "open question" which may be of interest to you, or if you teach, to your students. The question involves representing weighted voting games in a way that shows the Banzhaf power relations to the players.
Some ideas about weighted voting are given here, including the definitions and statement of the theorem of Alan Taylor and William Zwicker about when a voting game can be represented by a weighted voting game. The idea involves the trading of players between winning coalitions. Also, basic ideas about the apportionment model are developed. How should an integer number of seats, which must be assigned to claimants in integer amounts, be assigned based on the size of the claims put forward by the claimants?
This note has some information about apportionment models in classical (how many seats does a party get in a parliament based on the votes for the parties) as well as other settings. It is important that apportionment methods be viewed as fair which requires a way to judge whether or not one apportionment is better than another.
Three problems are posed involving payoffs in a two player game, where each player can choose two actings, and where the payoffs to the players add to zero. Our goal is to try to determine when a game of this kind is fair.
This note offers extensive references on apportionment problems as well as specific examples showing how different measures of optimality can be used to defend different apportionment methods. The approach developed here is whether or not the transfer of a seat from one state to another makes the measure smaller or bigger. A very different approach is a global optimization approach.
These are notes about games in extensive form which offer lots of ways to model conflict situations, including games with a dynamical quality (e.g. reactions of what player 2 can do to what player 1 has done). A brief discussion of Rheinhard Selten's work on extending that of John Nash is given.
Some notes about the problem of distributing a quantity E to claimants whose claims exceed E. Bankruptcy like problems arise in the distribution of water, or emergency funds, as well as problems concerning the funding of an amount E by collecting taxes from different income groups.
These situations give rise to many important problems in graph theory, operations research, and other parts of mathematics. For notes about what the key ideas are that these problems lead to, look at the following:
For the situations 5 and 6 which are not mentioned in the document above, the relevant notes are: Situation 5, Voronoi diagrams; Situation 6, Robbin's Theorem concerning when it is possible to orient a connected underdirected graph, so that the resulting digraph becomes strongly connected.
This election example introduces a notation due to Duncan Black (British political scientist) for expressing preferences of a "voter" in a situation where choices must be ranked by an individual voter. Choices listed towards the top are preferred by the voter.
This glossary offers a variety of terms that arise in the use of mathematics to study fairness questions. The terms are drawn from social choice theory (voting and elections), apportionment, and other domains.
One of the most remarkable theorems that mathematics has contributed as an insight into political science and economics is a result of Kenneth Arrow, who won the Nobel Prize for his work. The basic result is that for decision methods that produce rankings when there are three or more alternatives, there is no decision method which obeys a short list of reasonable "fairness" conditions.
An apportionment example using Webster, Jefferson's and Adam's Methods is worked out, Using the "divisor" approach to these methods. For relatively small house sizes one can usually do problems of this kind using "rank index" approaches to divisor methods.
Here is a sampler of fairness and equity problems for a general audience or to introduce a class in middle school or high school to some problems that lie within the domain of what mathematicians are studying that are related to fairness. Fairness Sampler
This essay discusses some mathematical models in political science. The public is accustomed to the effectiveness of using mathematics in physics, engineering, chemistry and biology. Howver, the fact that mathematics provides major insights into the social sciences is less appreciated.
g. Connection between facility location and statistics (This essay which supports the previous notes, deals with ideas connecting facility location problems and statistical concepts such as the mean, median, and mode. The audience is student in grades 6 and higher.)
Fairness Models
a. Voting and Elections (This is a primer about the history of mathematical insights into elections and voting systems.)
b. Apportionment I (This is the first of a two part essay about apportionment problems, such as deciding how many seats to give each US state in the House of Representatives based on the populations of the states.)
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You can find a broad list of topics dealing with fairness in this "syllabus" for a college level "humanities course" about fairness.
In the United States the President is not elected directly through popular vote but using the Electoral College. The Electoral College has 51 players who cast different numbers of votes, loosely proportional to the population of the regions the "electors" represent. Here is an example to show that a naive way of assigning weights to voters in a weighted voting game can result in "players" who have positive weight but no power!
General Modeling
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The Consortium for Mathematics and Its Applications has produced a wide array of excellent materials that deal with all aspects of Mathematical Modeling.
This includes many modules, and a journal (UMAP Journal) devoted to both research about mathematical modeling and educational issues related to mathematical modeling. Its "membership privilege" journal, published twice a year has lots of articles and materials about modeling.
A wide variety of mathematical modeling problems at various levels of difficulty and with a variety of levels are available on the web. There are also a variety of contests for high school students and college students about mathematical modeling. Some of these contests are run by COMAP and another by SIAM, the Society for Industrial and Applied Mathematics.
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Computer aided mathematics teaching
People
Brief description
The primary goal of this research group is to develop materials, technologies and methods to improve mathematics teaching and learning outcomes in the Aalto University School of Science. In order to support this goal, we research e-learning methodologies in mathematics. We are also actively involved in international collaboration and open source software development. For further information about our project and activities, see our portal, intmath.org.
Projects
We are taking part in Support Successful Student Mobility with MUMIE project, which is funded with support from the Lifelong Learning Programme of the European Union. More about the project:
Linda Havola: Improving the teaching of engineering mathematics: a research plan and work in-process report. In proceedings of the Joint International IGIP-SEFI Annual Conference 2010. Trnava, Slovakia, 2010.
pdf
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Why college or university students hate proofs in mathematics? (English)
J. Math. Stat. 5, No. 1, 32-41 (2009).
Summary: Problem Statement: A proof is a notoriously difficult mathematical concept for students. Empirical studies have shown that students emerge from proof-oriented courses such as high-school geometry, introduction to proof, complex and abstract algebra unable to construct anything beyond very trivial proofs. Furthermore, most university students do not know what constitutes a proof and cannot determine whether a purported proof is valid. A proof is a convincing method that demonstrates with generally accepted theorem that some mathematical statement is true and each proofs step must follow from previous proof steps and definition that have already been proved. To motivate students hating proofs and to help mathematics teachers, how a proof can be taught, we investigated in this study the idea of mathematical proofs. Approach: To tackle this issue, the modified Moore method and the researcher method called Z. Mbaïtiga method are introduced follow by two cases studies on proof of triple integral. Next a survey is conducted on fourth year college students on which of the proposed two cases study they understand easily or they like. Results: The result of the survey showed that more than 95\% of the responded students pointed out the proof that is done using details explanation of every theorem used in the proof construction, the case study2. Conclusion: From the result of this survey, we had learned that mathematics teachers have to be very careful about the selection of proofs to include when introducing topics and filtering out some details which can obscure important ideas and discourage students.
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HP33S Scientific Calculator
Calculated Industries
Accuracy, functionality, and dependability are three vital attributes that successful scientific projects require and the pocket-sized HP 33s scientific programmable calculator is designed to deliver them.
It also features 31KB user memory, the choice of RPN and algebraic data-entry modes, a powerful two-line display, and the timesaving HP Solve application making it an ideal choice in scientific calculators.
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This geometry book is written foremost for future and current middle school teachers, but is also designed for elementary and high school teachers. The book consists of ten seminars covering in a rigorous way the fundamental topics in school geometry, including all of the significant topics in high school geometry. The seminars are crafted to clarify and enhance understanding of the subject. Concepts in plane and solid geometry are carefully explained, and activities that teachers can use in their classrooms are emphasized. The book draws on the pictorial nature of geometry since that is what attracts students at every level to the subject. The book should give teachers a firm foundation on which to base their instruction in the elementary and middle grades. In addition, it should help teachers give their students a solid basis for the geometry that they will study in high school. The book is also intended to be a source for problems in geometry for enrichment programs such as Math Circles and Young Scholars.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
Readership
Undergraduate students interested in secondary education, particularly the teaching of geometry, and current middle school teachers teaching geometry.
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The important math your child
will work on in Don's books(note,
all of Don's materials come from work he has done with young people!):
Counting how many pieces of a size make the whole cake,
to name
a fraction of the cake in chapter
1, or cookie in chapter
2. This is a key idea which many students are not aware of and
causes
difficulty in all their math courses!
quadratic--by guessing!!, sum and
product of roots, by iteration many ways, by graphing, by quadratic
formula, using a calculator to hone in on the answer
cubic--by iteration, by computer
Iterating functions:
like 5 + x/2, if you start with 4 in
for x, what happens to the sequences?
to solve equations
to find compound interest
to find the square root of a number
to find the square root of the square
root..of a number
The use of computers and calculators: computer programs are in
almost
every chapter, with an appendix in the worksheet book on how to write
programs
to get infinite sequences and series. Don uses Derive to zoom in on a
curve to
get the slope of the tangent leading to the derivative. He uses
Mathematica to
show 100 iterations of a function. (See the use of computers page).
Probability: The area under the normal
curve is 1 and is related to probability
Trig functions: sine is used in finding the perimeter of
polygons
inscribed in a circle to get to Pi and is shown as an infinite series.
See also Trig
for young people.
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New Textbook
Related Products
Student's Solutions Manual for Elementary and Intermediate Algebra for College Students
Worksheets for Classroom or Lab Practice for Elementary and Intermediate Algebra for College Students
Summary
For freshman-level, one- or two- semester courses in Developmental Algebra. This dynamic new edition of this proven series adds two new titles and has cutting edge print and media resources. An emphasis on the practical applications of algebra motivates students and encourages them to see algebra as an important part of their daily lives. The student-friendly writing style uses short, clear sentences and easy-to-understand language, and the outstanding pedagogical program makes the material easy to follow and comprehend. The new editions place a stronger emphasis on problem solving, incorporating it as a theme throughout the texts. Angel's solid exercise sets have been expanded to include new Conceptual/Writing Exercises, Practice the Skills Exercises, Problem Solving Exercises, and Group Activities. A comprehensive supplements package includes a new Companion Website.
Table of Contents
Preface
xi
To The Student
xxiii
Real Numbers
1
(90)
Study Skills for Success in Mathematics
2
(7)
Problem Solving
9
(12)
Fractions
21
(12)
The Real Number System
33
(5)
Inequalities
38
(5)
Addition of Real Numbers
43
(8)
Subtraction of Real Numbers
51
(8)
Multiplication and Division of Real Numbers
59
(8)
Exponents, Parentheses, and the Order of Operations
67
(13)
Properties of the Real Number System
80
(11)
Summary
85
(2)
Review Exercises
87
(2)
Practice Test
89
(2)
Solving Linear Equations
91
(62)
Combining Like Terms
92
(9)
The Addition Property of Equality
101
(10)
The Multiplication Property of Equality
111
(9)
Solving Linear Equations with a Variable on Only One Side of the Equation
120
(7)
Solving Linear Equations with the Variable on Both Sides of the Equation
127
(8)
Ratios and Proportions
135
(18)
Summary
149
(1)
Review Exercises
150
(1)
Practice Test
151
(1)
Cumulative Review Test
152
(1)
Formulas and Applications of Algebra
153
(68)
Formulas
154
(13)
Changing Application Problems into Equations
167
(12)
Solving Application Problems
179
(16)
Geometric Problems
195
(5)
Motion and Mixture Problems
200
(21)
Summary
214
(1)
Review Exercises
215
(2)
Practice Test
217
(1)
Cumulative Review Test
218
(3)
Graphing Linear Equations
221
(50)
The Cartesian Coordinate System and Linear Equations in Two Variables
222
(10)
Graphing Linear Equations
232
(12)
Slope of a Line
244
(7)
Slope-Intercept and Point-Slope Forms of a Linear Equation
251
(20)
Summary
265
(1)
Review Exercises
266
(2)
Practice Test
268
(1)
Cumulative Review Test
269
(2)
Exponents and Polynomials
271
(58)
Exponents
272
(9)
Negative Exponents
281
(7)
Scientific Notation
288
(8)
Addition and Subtraction of Polynomials
296
(7)
Multiplication of Polynomials
303
(11)
Division of Polynomials
314
(15)
Summary
324
(1)
Review Exercises
325
(2)
Practice Test
327
(1)
Cumulative Review Test
328
(1)
Factoring
329
(58)
Factoring a Monomial from a Polynomial
330
(8)
Factoring by Grouping
338
(5)
Factoring Trinomials of the form ax2 + bx + c, a = 1
343
(9)
Factoring Trinomials of the form ax2 + bx + c, a = 1
352
(13)
Special Factoring Formulas and a General Review of Factoring
365
(7)
Solving Quadratic Equations Using Factoring
372
(15)
Summary
382
(1)
Review Exercises
383
(2)
Practice Test
385
(1)
Cumulative Review Test
385
(2)
Rational Expressions and Equations
387
(72)
Simplifying Rational Expressions
388
(7)
Multiplication and Division of Rational Expressions
395
(7)
Addition and Subtraction of Rational Expressions with a Common Denominator and Finding the Least Common Denominator
402
(8)
Addition and Subtraction of Rational Expressions
410
(9)
Complex Fractions
419
(5)
Solving Rational Equations
424
(9)
Rational Equations: Applications and Problem Solving
433
(11)
Variation
444
(15)
Summary
453
(1)
Review Exercises
453
(3)
Practice Test
456
(1)
Cumulative Review Test
457
(2)
Functions and Their Graphs
459
(70)
More on Graphs
460
(8)
Functions
468
(1)
Linear Functions
468
(29)
Slope, Modeling, and Linear Relationships
497
(17)
The Algebra of Functions
514
(15)
Summary
523
(1)
Review Exercises
524
(2)
Practice Test
526
(2)
Cumulative Review Test
528
(1)
Systems of Linear Equations
529
(68)
Solving Systems of Equations Graphically
530
(10)
Solving Systems of Equations by Substitution
540
(6)
Solving Systems of Equations by the Addition Method
546
(8)
Solving Systems of Linear Equations in Three Variables
554
(7)
Systems of Linear Equations: Applications and Problem Solving
561
(13)
Solving Systems of Equations Using Matrices
574
(8)
Solving Systems of Equations Using Determinants and Cramer's Rule
582
(15)
Summary
590
(2)
Review Exercises
592
(2)
Practice Test
594
(1)
Cumulative Review Test
595
(2)
Inequalities in One and Two Variables
597
(40)
Solving Linear Inequalities in One Variable
598
(15)
Solving Equations and Inequalities Containing Absolute Value
613
(10)
Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities
623
(14)
Summary
631
(1)
Review Exercises
631
(2)
Practice Test
633
(1)
Cumulative Review Test
634
(3)
Roots, Radicals, and Complex Numbers
637
(68)
Roots and Radicals
638
(8)
Rational Exponents
646
(8)
Multiplying and Simplifying Radicals
654
(8)
Dividing and Simplifying Radicals
662
(10)
Adding and Subtracting Radicals
672
(6)
Solving Radical Equations
678
(12)
Complex Numbers
690
(15)
Summary
698
(1)
Review Exercises
699
(3)
Practice Test
702
(1)
Cumulative Review Test
703
(2)
Quadratic Functions
705
(76)
Solving Quadratic Equations by Completing the Square
706
(10)
Solving Quadratic Equations by the Quadratic Formula
716
(14)
Quadratic Equations: Applications and Problem Solving
730
(7)
Factoring Expressions and Solving Equations that are Quadratic in Form
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Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix
Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between the two above said curves (the region should be clearly identifiable).
Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: dy / dx+ py = q, where p and q are functions of x or constant dx / dy + px = q, where p and q are functions of y or constant
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors. Scalar triple product of vectors.
Direction cosines and direction ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes. (iii) a line and a plane. Distance of a point from a plane.
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Tag Archives: algebra
Quadratics have become my new nadir, which is cheerier news than it sounds since it means I've kicked linear equations into obedient submission. For the first two and a half years of my teaching career, I felt good about quadratics because if nothing else, most kids remembered how to factor, and remembered that factors had something to do with zeros on the graph. Which was a big step up compared to what they retained of linear equations. But then, last year, I crackedlinearequations in a big way, which is great except now I just feel bad about quadratics, because as I develop as a teacher I realize the suckers are absurdly complicated and don't model very easily. The kids learn a lot, but at their level of ability I'd need to do two months to have them internalize quadratics the way most of them internalize linear equations. And I don't have two months. I just tell myself they still learn a lot. Consequently, I am relieved to see quadratics in the rear view as I move them onto the third of the models that define second year algebra (at least, as I teach it).
Exponential functions are awesome. First, they're absurdly simple compared to both lines and quadratics. Second, they model actual, honest to god, real life situations. I'm not a big teacher for "Hey, this is something you'll use again" but automobile depreciation or interest payments are, in fact, something they'll use again. Third, they provide a memorable and again, useful, reason to review (or learn for the first time) percentage increase and decrease. Finally, they present a situation in which any kid who has even somewhat grasped the course essentials can see hey: Given y, I can't solve for x. This leads beautifully and meaningfully into logarithms.
So like linear equations, I can kick off the unit with a modeling activity and get the kids moving easily into the math.
I begin with a brief lecture reminding them of the two previous models.
Then I review percentage increase and decrease. I am of two minds about this review. On the plus side, it's immediately relevant, easy to apply, and gives them a good reason to remember it long term. The downside: the kids never remember what I taught them when they get to the percentage problems. So I explain it up front, knowing that 90% of the kids will forget everything I said just 20 minutes later, when they get to the first percentage exponential increase.
So I explain it, go round the room asking "So, if I want to increase a number by 8%, what do I multiply it by, Jose?" "1 point…..8?" "Watch that leading zero!" "Oh, 1.08." "Right." Do that with five or six times, think everyone gets it, and set them to working on models. This is one side of the worksheet, crunched for space so I could "snip" it.
And sure enough, the kids work through the models, making great progress, and stop cold at the third one.
"I can't do this. How do you increase by a percentage?"
"Excuse me while I beat myself on the head with this whiteboard."
"What?"
"Nothing. Do you remember me just talking about percentages?"
"Yeah."
"Do you see it on the board there? All the stuff about turning it from two steps into one step, and why you need to do that?"
"Yeah."
"DO YOU SEE ANY POSSIBLE CONNECTION BETWEEN THAT CONVERSATION AND THIS PROBLEM?"
"Man, I don't see why you're so mean."
"Read what it says on the board. Right there. In red."
"Increase x by a%."
"Yes. Can you read problem 3 and tell me what you think might possibly qualify as x?"
"The population?"
"Yes. And do you see the value that might possibly qualify as a%?"
"Um." Long pause as the student stares at the problem, and finds the ONLY OTHER VALUE MENTIONED. "Twenty percent?"
"Indeed."
"Okay."
I repeat that four or five times to four or five groups and then, miracle of miracles, find a student with a full table of five values for the population problem. There is a god.
"Great."
"But I don't know how to find the equation for this one like I did the first two. This one isn't repeated multiplication. I had to take 20% of 250 and then add it….why are you hitting yourself on the head?"
"We need a function. We need an operation in which we can plug in x—do you have any thoughts on what x might be?"
"How many months?"
"How is it you know that, you smart child, and yet make me go through this torture? Yes. We need an operation that we can plug in the number of months (x) and get the population (y)."
"Right. But this is like three steps."
"And we need only one."
"Right."
"Wouldn't it be cool if there were a way to increase a number by a given percentage in just one step?"
"How do you do that?"
"LOOK AT THE BOARD!"
"Oh, is that what you were talking about? I was already doing the worksheet."
And still, the lesson is largely a success. Kids are absolutely freaked out at the cell growth caused just by doubling and yes, I bring up the million dollar mission example, but at the end of the lesson, not as part of it. Most of the kids correctly graph the models, although a few end up with lines that I correct. The flip side of the handout is a blank graph, which they use to take notes on the basic exponential growth model.
Total Amount = Initial Amount * Ratetime
Initial Amount > 0
Rate > 1
One thing I mull over—the book, and the state test, go through the exponential equation (basically, Initial Amount = 1), along with the transformation model (f(x) = ax-c +- k. I haven't focused on this in previous classes, because in my experience the kids don't even get tranformations of lines and quadratics. But I'm going to give it a try on Monday.
Anyway. Day 2 is exponential decay, but I start by going over percentage decrease. I am nothing if not optimistic.
"So if I take away a third of something, how much is left?"
Pause. Pause some more. Pause still more. I grab three whiteboard pens.
"Rhea, decrease these pens by a third." Rhea obediently takes one pen.
"Class, how much is left after she decreased the pens by 33%, or a third?"
"TWO!!!"
"Two……?" I wait. No. I sigh, and grab three more pens, getting the one back from Rhea as well.
"Paul, take away a third of these six pens." Paul takes two pens.
"Class, he's taken away 33% of the pens. How much is left?"
"FOUR!"
"AUUGGGGHHH!"
It all works out. Seriously. By the end of the exercise, most of the class is shouting back the correct answers as I ask "I take away 30%, how much is left?" 35%? 23%?" and the only mistakes they make are place errors—that is, 100-23 does not, in fact, equal 87.
The second day is always better, because it has slowly permeated their skulls that I'm serious about this percentage nonsense, that it has some relationship to the worksheet. So when they ask questions, it's more of the "could you run this whole percentage decrease by me again? If they take away a third, I have two thirds left? But what's two thirds as a decimal?" and trust me, this is a big step up for my blood pressure. Well, a step down. And they do the decay modeling and notes with no small degree of interest:
They have the model graph on the back, too, for exponential decay:
Total Amount = Initial Amount * Ratetime
Yes, it's the same equation, so what's different?
Initial Amount > 0
0 < Rate < 1
By day's end, they have registered the import of the realization that Estefania has 95 cents left after ten days, and they've figured out that Jose is right, that his car is worth more than Stan's after five years, which they managed by using an equation they built themselves, by golly, rather than decrease 25,000 by 5% 5 times.
You notice, of course, that I've spent most of this post talking about the percentage issue, something the kids learned were first taught back in middle school, than the exponential growth/decay functions, the actual new material. This should not come as a shock to regular readers.
This twitter debate between reformers Mike Petrilli and Rishawn Biddle is typical of reform debates about "rigor". Petrilli wants end of course exams to stop us teachers from pretending to teach a subject. Biddle wants more of the same, just shout louder and MANDATE instruction, particularly to those disenfranchised black and Hispanic youth who are being let down by lousy teachers with low expectations.
Both of them assume that the problem is ineffective teaching, that all us math teachers could actually teach percentages and fractions to all seventh graders if we were just smarter and better. Or maybe they just think we take the easy way out, that it'd be really really hard to teach the kids properly, and what the hell, we get paid no matter what and behind close doors it's easier to just go through the motions. Well, sure.
Petrilli's proposal, end-of-course exams, would trigger a bloodbath. People really don't seem to understand how I'd be all in favor of that, if the result were a rethinking of expectations. But of course, what would actually happen is that we'd end the end-of-course exams. That's what always happens whenever a state or district tries to enforce higher standards (cf Oklahoma and now Texas). And of course, that's what's going to happen with Common Core standards, assuming that anyone actually takes them seriously after the testing bloodbath this year. But I'd be all for end-of-course testing if reformers would accept responsibility for the 80% decrease in graduation rates among blacks and Hispanics who would never get past algebra I and understand, finally, that they believe in a myth.
But I digress. And I'm still going to like exponential functions, at least until I crack quadratics. Because you know what? The kids do make progress in understanding percentages, and they learn for the first time not only about exponential functions, but about asymptotes, as I explain Zeno's Paradox. I don't use Achilles and the tortoise as an example, but instead talk about how I could throw a stapler right at BTS's head and know that the stapler would never draw blood because it wouldn't reach his noggin, so I couldn't get fired. Or that I could walk to the door and never get there. I do get to the door, of course, and alas, the stapler would eventually crack BTS's skull. But even though we know that this is true, the tools for proving the paradox false, as opposed to demonstrating it, don't come around until calculus. They get a kick out of that.
If all that's not fun enough, I see genuine, honest-to-god intellectual curiosity among most students as they realize that they don't have the tools to isolate x in the equation 8 = 3x. That for all these years they've been getting along fine with addition/subtraction, multiplication/division, nth power/nth root, but none of those will work here. Which sets us up beautifully for both logs and a proper discussion of inverses, leading into inverse functions. Yes, their skills are still basic, but I can see the glimmering of understanding of the underlying concepts. If the damn state tests would just ask questions about those underlying concepts instead of demanding underlying concepts and advanced operations, I might even be able to get the kids to show that understanding.
And in writing up this essay, I am struck by the obvious solution to the percentage problem on day one: I need a worksheet. They fill it out, and not until they are done with that do I give them the worksheets on growth and decay. Naturally, this solution is again a lowering of expectations, a realization that a clear explanation on a blackboard that they can refer to isn't enough, that I need to give fifteen to seventeen year olds an activity so the information will sink in and they use the method right away without asking me to explain it all again group by group. But to hell with expectations. It will be much better for my bloodpressure tacos remind support pretty only better Instruction according andLast week, Education Week ran an article about a recent study from Southern Methodist University showing that students performed better on algebra word problems when the problems tapped into their interests. …The researchers surveyed a group of students, identified some general categories of students' interests (sports, music, art, video games, etc.), and then modified the word problems to align with those categories. So a problem about costs of of new home construction ($46.50/square foot) could be modified to be about a football game ($46.50/ticket) or the arts ($46.50/new yearbook). Researchers then randomly divided students into two groups, and they gave one group the regular problems while the other group of students received problems aligned to their interests.
The math was exactly the same, but the results weren't. Students with personalized problems solved them faster and more accurately (emphasis mine), with the biggest gains going to the students with the most difficulty with the mathematics. The gains from the treatment group of students (those who got the personalized problems) persisted even after the personalization treatment ended, suggesting that students didn't just do better solving the personalized problems, but they actually learned the math better.
Reich has it wrong. From the study:
Students in the experimental group who received personalization for Unit 6 had significantly higher performance within Unit 6, particularly on the most difficult concept in the unit, writing algebraic expressions (10% performance difference, p<.001). The effect of the treatment on expression-writing was significantly larger (p<.05) for students identified as struggling within the tutoring environment1 (22% performance difference). Performance differences favoring the experimental group for solving result and start unknowns did not reach significance (p=.089). In terms of overall efficiency, students in the experimental group obtained 1.88 correct answers per minute in Unit 6, while students in the control group obtained 1.56 correct answers per minute. Students in the experimental group also spent significantly less time (p<.01) writing algebraic expressions (8.6 second reduction). However, just because personalization made problems in Unit 6 easier for students to solve, does not necessary mean that students learned more from solving the personalized problems.
(bold emphasis mine)
and in the Significance section:
As a perceptual scaffold (Goldstone & Son, 2005), personalization allowed students to grasp the deeper, structural characteristics of story situations and then represent them symbolically, and retain this understanding with the support removed. This was evidenced by the transfer, performance, and efficiency effects being strongest for, or even limited to, algebraic expression-writing (even though other concepts, like solving start unknowns, were not near ceiling).
So the students who got personalized instruction did not demonstrate improved accuracy, at least to the same standard as they demonstrated improved ability to model.
I tweeted this as an observation and got into a mild debate with Michael Pershan, who runs a neat blog on math mistakes. Here's the result:
I'm like oooh, I got snarked at! My own private definition of math!
But I hate having conversations on Twitter, and I probably should have just written a blog entry anyway.
Here's my point:
Yes, personalizing the context enabled a greater degree of translation. But when did "translating word problems" become, as Michael Pershan puts it, "math"? Probably about 30 years old, back when we began trying to figure out why some kids weren't doing as well in math as others were. We started noticing that word problems gave kids more difficulty than straight equations, so we start focusing a lot of time and energy on helping students translate word problems into equations—and once the problems are in equation form, the kids can solve them, no sweat!
Except, in this study, that didn't happen. The kids did better at translating, but no better at solving. That strikes me as interesting, and clearly, the paper's author also found it relevant.
Pershan chastised me, a tad snootily, for saying the kids "didn't do better at math". Translating math IS math. He cited the Common Core standards showing the importance of data modeling. Well, yeah. Go find a grandma and teach her eggsucking. I teach modeling as a fundamental in my algebra classes. It makes sense that Pershan would do this; he's very much about the why and the how of math, and not as much about the what. Nothing wrong with this in a math teacher, and lord knows I do it as well.
But we shouldn't confuse the distinction between teaching math and doing it. So I asked the following hypothetical: Suppose you have two groups of kids given a test on word problems. Group 1 translates each problem impeccably into an equation that is then solved incorrectly. Group 2 doesn't bother with the equations but gives the correct answer to each problem.
Which group would you say was "better at math"?
I mean, really. Think like a real person, instead of a math teacher.
Many math teachers have forgotten that for most people, the point of math is to get the answer. Getting the answer used to be enough for math teachers, too, until kids stopped getting the answer with any reliability. Then we started pretending that the process was more important than the product. Progressives do this all the time: if you can't explain how you did it, kid, you didn't really do it. I know a number of math teachers who will give a higher grade to a student who shows his work and "thinking", even if the answer is completely inaccurate, and give zero credit to a correct answer by a student who did the work in his head.
Not that any of this matters, really. Reich got it wrong. No big deal. The author of the study did not. She understood the difference between translating a word problem into an equation and getting the correct answer.
But Pershan's objection—and, for that matter, the Common Core standards themselves—shows how far we've gone down the path of explaining failure over the past 30-40 years. We've moved from not caring how they defined the problem to grading them on how they defined the problem to creating standards so that now they are evaluated solely on how they define the problem. It's crazy.
End rant.
Remember, though, we're talking about the lowest ability kids here. Do they need models, or do they need to know how to find the right answer?
(okay, a brief note. Early in his writeup, Nebus (Joe? Joseph?) writes: "It's a method for factoring quadratic expressions into binomial expressions, and I must admit, it's not very good. It's cumbersome and totally useless once one knows the quadratic equation."
Many, many math teachers have expelled much breath on the uselessness of factoring, as the skill is completely nullified by the quadratic formula (which I think what he means here). But when they make this comment, they are thinking as mathematicians, not as teachers. Mathematicians work with math to solve problems. Teachers teach math so their kids can demonstrate their knowledge on tests*—not just state tests, but college admissions tests and placement tests. And on these tests, the questions are designed for either factoring or the quadratic formula—and far more the former than the latter. All students must learn to factor trinomials if they are to escape remediation. The quadratic formula is optional. And if the test pragmatism isn't enough of an argument, please know that students with limited integer operations skills will do better with factoring than the formula because they rarely have squares memorized and please, please believe me when I say that they will ALWAYS subtract 4ac from b squared, even if c is negative. End brief note.)
There are teachers who think this is a science, and teachers who know it's an art based largely on the audience. And which kind of teacher you are is a religion, or an expression of personality (which I often think is the same thing).
So when I say that the method Nebus describes sounds extremely convoluted, I am simply a Jehovah's Witness expressing doubt about the utility of the Amish rumspringa.
But many math teachers aren't even aware of the box/diamond method, and many others who do use it don't teach it in a fully integrated manner. So for the teachers of the artist mindset looking to find the right method for certain audiences, here's an outline of the method.
I got the approach from CPM. I don't know if it originated with CPM, so apologies if the original idea goes back earlier. CPM's curriculum is insanely irritating: text heavy, lousy examples and insufficient practice. But in many cases, its approach to a topic provides a beautiful, fully integrated, and consistent framework that I steal without shame.
Factoring out common terms
I always introduce the generic rectangle when introducing or reviewing simple factoring (pulling out common terms). The area model uses the fact that the rectangle's area is both the product of the length and width and the sum of the individual areas. You break up the side of the rectangle into as many different segments as there are terms.
So 8x + 18 is the sum of two areas, both created by a product of length and width. One side is used for both areas, so it must be a factor common to both areas. In other words, what is the greatest common factor of both terms?
Once you find the GCF, work backwards. What do I multiply by 2 in order to get 8x? Most students do well on this, but if they struggle, I show them how to divide in order to find the answer.
I don't stress its use here, as I do during binomial multiplication; I just want students to be familiar with my use of it. At the same time, I always find a few students who struggle with factoring common terms and find the approach helpful.
Binomial Multiplication
I do not mention FOIL, although most of my students have learned it at one time or another. While I don't require my students to use any particular method for tests, I require them to use the area model method for binomial multiplication at least for a day or two so they understand the underpinnings of the factoring method.
So obviously, binomial multiplication is the opposite of factoring; the terms go on the outside of the box and generate the individual areas. (x+2) is a segment of length of x and 2, (x+3) a segment of x and 3. I always point out that the lengths don't need to be accurate or drawn to scale.
I demonstrate this method several times, up front. I explain the area concept again and how multiplication of length times width for each smaller rectangle is the same as the area of the larger rectangle. I don't really expect my students to be able to repeat it back to me. What I expect, or hope, they will think is "Oh, okay, that makes sense". Because from this point on, when they think of this method, I want them to remember that the method made sense to them, even if they don't remember the specifics. That's also why I don't write any of this down—most of my students will toss any documentation, anyway. I work a variety of examples (at this point, a=1), picking students to walk through the process.
The kids have a handout with 20-30 problems (this is one of the few topics that kuta software doesn't have a good handout for), but I don't have my usual handout online. The original problems would all be a=1, b and c of all signs, because I want them to work dozens of problems and see the pattern, if they are able to. Then, on day 2, 3, and 4, I introduce difference of two squares (what happens to b?), a>1, and 2×3 or 3×3 polynomial multiplication—which works really well with this model, as the kids just make a bigger rectangle.
I wish I could say that this method eliminates the problem of (x+2)(x+3) = x2 + 6. Alas. However, when a student makes the mistake and I scowl and draw the rectangle, with no other explanation, 9 out of 10 kids making the mistake go "Oh, yeah." That's the win, such as it is.
Factoring Trinomials
So after a week or so of multiplication, I point out something interesting about the completed rectangle:
This is particularly interesting when we consider the two "middle" terms that add up to bx. We now know that they add to bx and multiply to the same product as ax2 and c.
Interesting, yes, but also useful. I remind the kids that distribution is the inverse of factoring, that distribution converts a product into a sum, while factoring turns a sum into a product. So if they are faced with a quadratic equation in ax2 + bx + c—say, for example, x2 + 9x + 14—how could they turn this sum back into a product?
I ask the kids, if I'd multiplied two binomials to get x2 + 9x + 14, what would have been in the box?
Factoring trinomials is the task of finding the numbers for the other half of the rectangle.
And thanks to the properties of the generic rectangle, we know that the terms we are looking for add to 9x and multiply to 14x2, the product of the first kittycorner.
So we use the "diamond" as a visual tool to help find those terms.
No matter what method a teacher chooses to teach, factoring comes down to that question: What do I multiply to get ac that I add to get b?
I teach the students to write out the factors in pairs, starting with 1 and the number itself (otherwise they tend to forget) and working up from there. Remember that I teach students who will often have a tough time remembering all the factors of 24, and pause on each term to remember the pair.
So once you find the terms that meet the requirements, you put them in the box. It doesn't matter which goes where. I repeat that phrase a lot. I sometimes wonder if I should create a rule for where the terms go, just so I won't get the question any more.
It's worth stressing to your students that, while you've found the missing terms, you still have one more step! They'll still forget, and this will bite them back when they start factoring trinomials in which a>1.
The last step involves finding the GCF for each row. This is where I get the payoff for introducing "single row" factoring much earlier. The students are familiar with the process; they've seen me explain that the outside terms are the GCF for a month or more, even if they didn't use it themselves.
Again, I work five or six problems with the class as a group each day. The kids have a page of 20-30 problems they work through; if they finish one page, I give them another with more complex problems. Anyone who can do the work peels off from the class discussion and works independently from the beginning, the rest are "released" after the class discussion. I put worked examples all over the whiteboards to give them models to follow. Many of my struggling students don't move past a=1. Some of the weakest will only be reasonable competent at c>0 in the first go-round and struggle with finding the difference of two terms for a while. So over the next two-three days, the kids work on factoring at their own pace. The strongest kids are working a>1 by the last day (and their third page of problems), and working problems like x2 – 9x = 10, learning to set it equal to 0 and factor.
Here's an example with c<0:
Here's a>1—and this, by the way, is where anyone can benefit from the generic rectangle. Any other method of factoring a>1 trinomials is a pain in comparison:
I return to factoring throughout the year. Every so often I'll declare it time to build on existing skills, so kids who had just gotten competent at c>0 can get more practice time on c1, and then the strongest kids start to identify patterns—identifying perfect squares, difference of two squares, and so on.
As time goes on, I give fewer worked examples and just the general outline below, to see how they do at moving from general to specific:
Next Steps
I have traditionally gone from this to completing the square and quadratic formula, then onto graphing parabolas. I am going to reverse these two topics this year. Teach factoring trinomials, then graph parabolas. Get that going well, and then move onto completing the square, quadratic formula, and then graphing those cases. See how that goes.
Finally, I can't stress this enough: a quarter or more of my algebra classes are low ability kids, so if you're thinking Jesus, two weeks or more for multiplying and factoring quadratics? then you aren't teaching low ability kids or you're just ignoring the fact that they're flunking your class. My top kids are doing in depth work on the topic or, in some cases, moving onto another topic entirely.
I've been getting some people lately asking, or complaining, that "low ability" is vague. I'm sorry, but it's not. Potter Stewart was right: you know it when you see it. If you want a specific metric, it's a kid with cognitive abilities measured at the 50th percentile (say, IQ from 95-105, but that's a guess). In other words, kids that are perfectly functional in the real world, but simply don't have the interest or ability for advanced math. Kids with cognitive abilities any lower than that aren't, as a rule, going to be able to even fake it in algebra, much less anything past that. There are always exceptions.
It's the delusion of eduformers and progressives, one and all, that if teachers find the right approach, a low ability kid is transformed into a competent high ability kid. In reality, success in teaching low ability kids comes when they start to feel a sense of competence at some level of math. When a kid goes from staring blankly at a trinomial to thinking "Oh, yeah" when I draw the rectangle, that's a big goddamn win. I believe a lot of kids in this category could learn specific high level math in the context of a concrete task, although I have no evidence of this. But we'd have to sort kids into different groups and sorting's just one big no-no.
However, this method is helpful for kids of all abilities. High ability kids get a real kick out of seeing the link between the area model and the algebra, and I've rarely met a kid who didn't appreciate the utility of this method for a>1.
I don't have a handout per se for this whole method; what I've just laid out is 8-10 days of practice, followed by days interspersed here and there throughout the year. However, if I get a kid who comes in late, or who wants a specific tutorial, I have a document that I really need to rework, which is why I spent some time creating images for this writeup. But remember, all of this is religion and on factoring, I'm in a state of epistemic closure. Convert or live life as a heretic. I was going to say "Die, infidel", but really, the current insanity in the mideast takes all the hyperbole out of that statement and thus all the fun.
*If you are a mathematician who is also a teacher, stop hyperventilating. It's true. You know it is. Embrace the reality we live of understanding customized hybrid something
I had an interview for a middle school math position (why they called me, I dunno, I'm sure I was just filler), and had to bring a sample lesson plan.
I always create worksheets when asked to bring lesson plans, since it gives a better idea of how I teach. I read the standards and came up with this cartoon as a starting point:
I use buying decisions all the time in math classes, because the phrase "Pretend that it's your money" has a near-magical effect on kids who would otherwise ignore word problems.
So I'd kick off a discussion by asking the kids to identify the difference between the two questions. Most kids of any ability will realize that the girl's question has one answer, while the boy's question has several.
From there, I drill down to the boy's question. (click on the image for a slightly better view). Don't get too quibbly on the definitions—math purists are insanely annoying.
Now, here's the thing: this is not a lesson on functions. This is a lesson on variables and generating tables. It's seventh grade math, not pre-algebra or algebra.
But in order to get kids to think about variables and tables, I find it's helpful to start with a familiar situation and lead it back to math. In order to explain the difference between the girl's question and the boy's question, I have to introduce functions, which leads us to variables.
In my opinion, teachers don't spend enough time leading kids through the process of identifying the definition of the unknowns (because trust me, if you say "identify the unknowns", half the class says "Three Cheeseburgers!" and suddenly you're in a Bill Murray skit.) and generating a table of solutions to the problem. Teachers see these as obvious, as mere activities that we use to solve problems. But low to mid ability kids see nothing obvious about defining variables and generating tables. I've been slowly realizing this over time, and it crystallized into my data modelling unit in my Algebra II class. This activity, which I drummed up for an interview (but will undoubtedly use at some point), was my attempt to move it back to a pre-algebra state.
Many teachers who only work with high ability kids would see this handout as ludicrously easy and in fact, I'd hand this worksheet to my top seventh graders and tell them to go it alone, while I talked the rest of the class through. Top kids should figure this out fairly quickly, and on day 2, when the rest of my kids are still practicing simple situations, they'd be on to more advanced scenarios (giving two points and figuring out the rule, giving them a table, and graphing).
But many teachers who work with lower ability kids work hard to find meaningful questions and yet miss the mark by making the question too complex. I was talking to one of my curriculum instructors, and he asked me about a lesson he was planning to give to middle school students in a low income district. He wanted to give kids fractions like 7/20, 8/25, 9/30 or 8/30, 9/35, 4/15 and have them just ballpark whether the fraction was greater or less than the obvious value nearby. So is 7/20 greater than or less than 1/3? and so on.
I told him that he was dramatically overestimating their ability, and recommended he start by doing the same thing with halves. So a list of numbers: 2/5, 9/17, 5/11, and so on. Greater than or less than one half? The kids who can do that quickly, move onto numbers around 1/4. Only after a few iterations should he give the strongest kids the exercise he was thinking of, which most of the kids wouldn't be able to do, full stop. But he could build the weaker kids ability and fluency simply by getting them to think about greater or less than one half. I advised starting with the much simpler exercise. If I was wrong, he could have the more challenging activity ready to follow up, no harm done.
He took my advice and reported back. Most of the kids never got beyond the over/under 1/2, finding it challenging and meaningful. Most of the teachers he was working with had been sure the activity would be too easy.
I do think it's important to set open questions for all kids, not just the top students, to let them tussle with before you settle into working problems for practice. However, the questions must be tailored to student ability, and math teachers dramatically overestimate the ability of their kids to work with these questions in a meaningful manner.
It gets old to be told that acknowledgement of low cognitive ability is "setting expectations low". One of the comments on my "myth" post was "One question that I have is the extent to which students are playing Whack-a-Mole in order to get easy classes for a week or two." My original response was rather rude, so I changed it to a simple "No". It boggles my mind that anyone would think kids are faking it.
At this point, some teachers are remembering the time they taught their algebra intervention kids about logarithms, or complex numbers, or trigonometry cycles. I'm not saying you can't teach it. They just won't remember it. It will be as if they've never been taught explanation
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