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Author: Steve Awodey Publisher: Oxford University Press ISBN: 0199587361 Size: 32.40 MB Format: PDF, ePub View: 7049 DownloadRead Online
Useful for self-study and as a course text, the book includes all basic definitions and theorems (with full proofs), as well as numerous examples and exercises.
Author: F. William Lawvere Publisher: Cambridge University Press ISBN: 0521894859 Size: 72.99 MB Format: PDF, Docs View: 1922 DownloadRead Online
This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. | 677.169 | 1 |
Can you please be more detailed as to what sort of service you are expecting to get. Do you want to understand the principles and work on your assignments on your own or do you want a tool that would offer you a step-by-step answer for your math problems?
Algebra Helper indeed is a very good software to help you learn math, without having to go to school. You won't just get the answer to the question but the entire solution as well, that's how concepts are built. And to do well in math, it's important to have strong concepts. I would advise you to use this software if you want to finish your project on time | 677.169 | 1 |
The Scientific Calculator
What is a scientific calculator?
A scientific calculator contains functions that are used routinely
by scientists and engineers. A good scientific calculator must
be very precise and it must handle the huge dynamic range of numbers
that are encountered in our understanding of science. Scientific
calculators must often be used to extend our ability to make judgements
based upon observations. This means that they must provide functions
that help us analyze data as well as functions that simply calculate
numbers.
Basic functions of a scientific
calculator
A four function calculator can perform addition, subtraction,
multiplication and division. These four basic functions represent
the heart of any calculator. A Scientific calculator uses these
four functions to perform higher level functions. For the purposes
of this work, I arbitrarily segment the operations of a scientific
calculator into the following categories:
Accuracy and Precision
Scientific electronic calculators are accurate. The precision
of a calculator should be measured, rather than taking it for
granted. Problems in accuracy can arise when repeated calculations
are made on a single result. The nature of computational environment
will determine the ultimate accuracy of long calculation. Precision
is very important when doing division. Let's take a look at a
quick example.
x = 1/3/3/3/3/3/3*3*3*3*3*3*3*3
by definition, this value should equal 1. We have divided 1
by 3, 6 times and then multiply it by 3, six times. The problem
with division by three stems from the fact that it is a repeating
fraction that can never be represented with absolute accuracy.
Multiple operations begin to accumulate errors.
The example cited above could also be expressed as
x = (1/3)^6 * (3)^6
or
x = (3)^(-6) * (3)^6
or
x = 10 ^(-6 *log (3) + 6 * log (3))
Each of these four expressions should yield exactly the same
value (1 ). The mechanisms used by the calculator to determine
each result is different.
The first expression just uses basic division and multiplication
to perform the calculation. The second expression uses division,
exponentiation , and multiplication. The third expression utilizes
negative signed exponentiation, the final expression uses the
logarithm to find a solution. If we carry out the calculations
on a modern scientific calculator such as the HP32s and an older
HP-97 we get the following table of errors:
equation
HP-32sII
HP-97
1
-1 X 10^-12
-1 X 10^-10
2
-9 X 10 ^-12
-4 X 10^ -10
3
-2 X 10^-12
0
4
0
0
table of errors for the sample calculation
From this short analysis, we see that the newer HP32sII has
2 more digits of accuracy than the HP-97.The newer calculator
is 100X more precise than the older machine. Note that in the
third equation, it appears that the older machine is more accurate.
This could be the result of truncation in the exponent. The number
of digits of in a calculation are often less consequential then
the method used to round a number prior to calculation. These
errors begin to manifest themselves in division calculations.
Numeric Representations
to the User
The wide dynamic range of the numbers used in scientific calculations
leads to a requirement to represent numbers in terms of a mantissa
and an exponent. The display of numbers on a calculator also requires
some care in formatting. The formatting of a number was a particular
challenge to designers of early calculators. The native language
of the user often dictated the use of a comma where a decimal
point might be used. This is difference is found in the differences
in notation between English and German. The english value 1,234,567.89
is represented as 1.234.567,89. The differences in separators
and other notation led to some difficulties in design of the machines.
Users of HP calculators had a the ability to "fix" the
display decimal point and select the presentation method to the
display. Typical notations were SCI for scientific notation, ENG
for engineering notation and FIX for fixed decimal point . Scientific
notation uses a format of x.xxx X10^yy . Fixed point notation
rounds the display to the required number of decimal points. Engineering
Notation always presented an exponent that was divisible by 3.
Entering Numbers and Keyboard
Functions
There are two basic logical methods that can be used to describe
the operation and entry of numbers on a calculator: Algebraic
and Reverse Polish Notation (RPN). Each method has it's advantages
and disadvantages. Many engineers perfer RPN machines because
of their inherent clarity of operation. Others prefer Algebraic
entry because of the similarity to hand calculation. Each method
will be detailed a little later. First we'll explain a bit about
the internal construction of a calculator. The calculator can
be thought of like a house: there is an outside with it's facade
and there is an inside which is unique to the design of the calculator.
The keyboard of a caculator is like a door and the display of
a calculator is like a window. While calculators differ in details,
they all need a keyboard and a display. Entering a number into
a calculator is just like entering a house through a door. Once
you are inside, there are numerous directions you can go. When
a number is pressed on the keyboard, it is displayed on a display
to indicate that a choice has been made and the nature of that
choice. The entry of numbers on calculator keyboard is based upon
the Arabic Notion of reading a number from left to write.
Keyboard entry is the most basic algorithm found in a calculator.
As each succeeding number is entered, the value in the keyboard
register is multipled by 10 and the new number added to the result.
When the decimal point is pressed, the value that is entered is
divided by 10 to the power of the decimal point and added to the
result. Each press of a key must be examined for validity and
pupose. The left to right entry of the number system is natural
for the human but someone complex at the machine level.
The keyboard can be divided into three distinct functional
groups: Numeric, Operation, and Function. This distinct division
can be seen on many early calculators. The HP-35 is a good example
As a general rule, function keys affect the value in the display,
operation keys manipulate memory, and numeric keys enter numbers
into the display.
Calculators utilize the concept of a register. A register is
a place to store a number that will be affected by an immediate
operation. Calculators also use standard memory registers that
are used to to "hold" data or programs that will be
used later in the process. The separation between an RPN logic
calculator and an Algebraic machine is defined by a subtle difference
between actions of the operation keys.
The Algebraic machine uses the operation key to move the number
from the display register to an operand register. At the same
time, the implied function of the operation key is "tagged"
to this register. A new number is then entered in the display
register. If the "=" key or another numeric operation
key is pressed, the calculation is performed and the result is
displayed. If another operation key such as a "+" has
been pressed the result is placed back in the operation register.
Entry of another number and a depression of the "="
key will process the result and display it.
The RPN machine utilizes the concept of a "stack"
of registers. These machines utilize a single key to "push"
a value up the stack. An operation key such as a "+"
causes the first value to be "popped" and added to the
displayed value. Early calculators, like the HP35 had a four level
stack, labled x,y,z,t . Data could be pushed up the stack and
used later on in the calculation. For example the following example
may be used:
Solve X = 1 + cos (3.8 * Pi)
1 <enter> 3.8 <enter> PI, X, cos, +
An algebraic machine would need the following key strokes
1 + ( 3.8 X PI) cos =
The number of keystrokes is the same , but the ordering of
operations is different. Debate rages about which system is better
for everyday use. Algebraic notation gets very messy when dealing
with functions of two variables. For instance and operation such
as Rectangular to polar conversion. Entering the data is not really
a problem, the function key is treated as an operation key. That
is to say we enter the x value, press->P enter the y value
and then press "=". The problem is with the second value.
How do we get that. The answer is different on each machine. RPN
machines always have a stack and the ability to swap the x and
y registers. This allows convient storage and display of data.
Some early Algebraic machines actually used memory registers to
store the answers to multivariable functions.
Sexagisimal Numbers
Sexagisimal numbers are useful in time and angle calculations.
The unique nature of these numbers is that they are associated
with measures that have audifferent modulos based upon the position
in the numeric representation. For instance, a common representation
for time might appear as 11:33:14.5 . The leading digits are hours
and may be expressed modulo 12 or 24. The next set of numbers
is minutes and this modulo 60 and carries over into the hours.
The next set of digits is seconds which is also modulo 60, but
it has a modulo ten fractional part. The easiest way to handle
calculations involving time is to "cast" the time into
absolute seconds from a given date. This avoids all modulo problems
during the mathematical operations. At the end of the calculation,
the time is recast back to the HH:MM:SS.XX format.
Fractions
This section is under revision.....
Algorithms and Methods used in
Calculators
The Scientific calculator must work with numbers that extend
over a wide numeric range with high precsion. In addition to this,
a numeric entry is often subjected to a number of transformations
before arriving at an answer. Early calculators had very limited
space for programs so it was necessary to arrive at a balance
of accurate algorithms that occupied minimum space. In many instances
the result of one key press was maintained for use in the next
function. This was clearly seen in the early Monroe calculators.
The "2nd function" key truly acted on the result from
the first press of the key AND it was pressed after the function
key. This is in contrast to a modern calculator where the 2nd
function key alters the meaning of the operation key. The older
Monroe or Compucorp machines would calculate a SINE. Depressing
the second function key would then calculate the COSINE. A modern
machine would generally calculate the ARC SINE function. which
is a completely different algorithm. As memory and calculation
power increased, the need for cleaverness and ingenuity in algorithm
development for basic calculation decreased.
The basis of any good scientific calculator starts with an
excellent multiply and add capability. Without these two functions,
no other functions can be derived. At face value, it would seem
that this should be trivial, but it is important that most early
calculators (as well as many modern machines) worked in base ten,
BCD (Binary Coded Decimal) arithmetic. This eliminates some of
the simpler binary shifting techniques to perform multiplication.
The challenge to the earlier designer of calculator algorithms
was to utitlize those multiply and additions to drive the solution
to the rest of the functions.
The Elementary functions
For the purposes of this work, we'll define the elementary
functions as :
inverse (1/x)
sine / arc sine
cos/ arc cos
tan/ arc tan
exponential / logs (base ten and natural)
hyperbolic trig functions (sinh, cosh, etc.
with the possible exception of the hyperbolic functions, the
early scientific calculator could be expected to include all those
functions. While the need for these functions was well defined,
the method of calculation given minimal memory and minimal cycle
times was not so clear. Let's look at some methods and considerations:
Range Reduction, Final Rounding
Consider the need to calculate the SINE of an arbitray angle.
The function is completely defined over the range of 4pi and it
repeats over that interval for an infinite period. If we have
the ability to limit all entries to a range of 0 to 4Pi, then
we need only concentrate on the numeric solution within this range.
This process is called "RANGE REDUCTION" and it is a
fundamental element of all scientific calculators and is often
responsible for large errors . In some cases the errors are real
but quite small. Try taking the sin(x) where x = 10^22. This number
is interesting because it can be exactly represented in IEEE-754
double precision format. The exact result is:
-0.8522008497671888017727.....
My HP49G calculator reports
-0.852200849762
The difference is in 5 parts in the 12 decimal place. Hardly
something to worry about, but it does illustrate a practical limitiation
in modern calculator technology. Of course, having a computer
isn't necessarily a benefit. If we choose to calculate this value
using single precision (IEEE-754) mathematics, the resultant answer
would be:
-0.73408......
This relatively huge error(9 parts in the second decimal place)
comes from the lack of precision in the input numeric representation.
Polynomial methods
The polynomial is the basic algorithm used in most early scientific
calculators. Accurate polynomial descriptions to a function over
a rather large interval can require polynomials of high degrees.
For instance, approximating the function ln(1+x) over the range
[-1/2,1/2] with an error less than 10E-8 will require a polynomial
of degree 12. Note that this requires a transformation or range
reduction first.
Table Based Methods
The use of tables and interpolation can produce excellent results
but the memory requirements generally limited the use of tables
in most calulators. The advantages of tables include speed and
accuracy, but this is of no use if there wasn't enough memory.
Shift and Add algorithms
The CORDIC Algorithm
The CORDIC algorithm was the basis of the handheld revolution.
The key feature of the CORDIC algorithm is that it can be utilized
in the calculation of all the elementary functions. | 677.169 | 1 |
Aqa gcse mathematics homework book answers
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Paperback | November 3, 2016
Pricing and Purchase Info
about
Essential MATLAB for Engineers and Scientists, Sixth Edition,provides a concise, balanced overview of MATLAB's functionality that facilitates independent learning, with coverage of both the fundamentals and applications. The essentials of MATLAB are illustrated throughout, featuring complete coverage of the software's windows and menus. Program design and algorithm development are presented clearly and intuitively, along with many examples from a wide range of familiar scientific and engineering areas.
This updated edition includes the latest MATLAB versions through 2016a, and is an ideal book for a first course on MATLAB, or for an engineering problem-solving course using MATLAB, as well as a self-learning tutorial for professionals and students expected to learn and apply MATLAB.
Updated to include all the newer features through MATLAB R2016a
Includes new chapter on complex variables analysis
Presents a comparison of execution time between compiled and un-compiled code that includes examples
Describes the new H2 graphics features
About The Author
Daniel Valentine is a Professor of Mechanical and Aeronautical Engineering at Clarkson University and Affiliate Director of the Clarkson Space Grant Program which is part of the New York NASA Space Grant Consortium. This program has provided support for undergraduate research appointments, and for graduate students. He is currently inv... | 677.169 | 1 |
1987 ap english essay
Arithmetic Progression, Geometric Progression and Harmonic Progression are interrelated concepts and they are also one of the most difficult topics in Quantitative Aptitude section of Common Admission Test, CAT. We will discuss them one by one.
Arithmetic Progression (AP) The progression of the form: a, a + d, a + 2d, a + 3d … is known as an AP with first term = a,and common difference = d.
In an AP a, a + d, a + 2d, a + 3d, …, we have:
(i) nth term, T n = a + (n – 1)d (ii) Sum to n terms, where l is the last term. (iii) If a, b, c are in AP, then b is called with arithmetic mean (AM) between a andc. In this case, b = (a + c). | 677.169 | 1 |
The National Mathematical Centre (NMC) was established by ACT, CAP N58, 2004, to among others, train and develop high level of personnel in Mathematical Sciences, create a resource centre to serve Read more...
The Registry Department is the life-line of the workings of the Centre and it provides various supportive administrative services for the Centre. It is responsible for the day-to-day running of the Organization's logistics. Read more...
The study of the theory, experimentation, and engineering that form the basis for the design and use of computers.
"
Welcome to the official website of the National Mathematical Centre!
You are welcome to National Mathematical Centre, an International Centre of Excellence in Mathematics and Mathematical Sciences (Mathematics, Mathematical Sciences Education, Statistics, Theoretical Physics and Computer Science).Read more...
Prof. S. OnahDirector/Chief Executive
"
A new paradigm of Educational development is fast emerging and traditional societies are being transformed into knowledge societies. Information technology is key to achieving our corporate objectives.
Mrs Bamidele OluchiHead, Computer Science
Events Workshops & Conferences
Olympiads Awards Ceremony
The DG & NMC Staff presenting awards to students.
NMC Staff and Students Attend Conference
A group of NMC Staff and Students at a Conference.
Workshop on Computer Driven Data Analysis
Computer driven data analysis workshop with HOD's.
DG's Speech
The Registry Department is the life-line of the workings of the Centre and it provides various supportive administrative services for the Centre. It is responsible for the day-to-day running of the Organization's logistics. Read more...
NMC has produced several instructional materials over the years. Some of the NMC products, which are the basic tools for the implementation of its Mathematics Improvement Programme (MIP), consist of.
Teaching Modules for Teachers - Primary 1,2,3,4,5,6
1
Workbooks for pupils - Primary 1,2,3,4,5,6
2
Mathematical Games for Secondary Schools
3
The Whiz-Teacher (a device designed for ICT based teaching methods)
4
OUR MISSION
To develop appropriate initiative and resources of international standing for the re-awakening and sustaining interest in the mathematical sciences and their application to life and by so doing produce specialists in the Mathematical Sciences at all levels of our educational system.
OUR VISION
To become a world class Centre of excellence for research and training in the Mathematical Sciences capable of promoting the development and socioeconomic impact of Mathematical Sciences in Nigeria, as well as using mathematical Sciences to solve important scientific and technological problems. | 677.169 | 1 |
Sharp Scientific and statistical calculator The Sharp EL531XHBPK Scientific calculator is the latest edition to our popular line-up of scientific calculators for high school and university students ...
TI-84 Plus functionality and design that features a crisp colour screen Visualise concepts clearly and make faster, stronger connections between equations, data, and graphs in full colour ; Enhanced...
Since the introduction of the first electronic desktop calculator, sharp has led the way in the field of calculator technology. A history of superior engineering, advanced functions, and innovative te...
Since the introduction of the first electronic desktop calculator, sharp has led the way in the field of calculator technology. A history of superior engineering, advanced functions, and innovative te... | 677.169 | 1 |
This lesson is designed for high school, college-level, or adult learners of English as a second language.
This is a complete lesson which introduces students to how to use basic calculus expressions in spoken English. It includes among other things: relevant vocabulary, a fun hidden-picture activ | 677.169 | 1 |
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Every textbook answer. I needed this so badFree videos from MIT for high schoolers & teachers
Elementary Algebra is generalized form of arithmetic. It provides a language to represent problems and functions. Algebraic thinking is also one of the first forms of abstract thinking that students develop in mathematics.
The National Academies Press (NAP) was created by the National Academy of Sciences to publish the reports of the National Academies of Sciences, Engineering.
AP Calculus Volumes by Cross Sections Introduction, Lesson, Practice
A complete lesson that completely covers the calculation of volumes by cross sections. Most calculus textbooks ignore this topic, or jump straight to very difficult examples.This lesson begins with a review of geometry. Most calculus students have a difficult time writing an expression for the area of a cross section. | 677.169 | 1 |
Technical Mathematics, 6th Edition
Description
This textbook has been in constant use since 1980, and this edition represents the first major revision of this text since the second edition. It was time to select, make hard choices of material, polish, refine, and fill in where needed. Much has been rewritten to be even cleaner and clearer, new features have been introduced, and some peripheral topics have been removed.
The authors continue to provide real-world, technical applications that promote intuitive reader learning. Numerous fully worked examples and boxed and numbered formulas give students the essential practice they need to learn mathematics. Computer projects are given when appropriate, including BASIC, spreadsheets, computer algebra systems, and computer-assisted drafting. The graphing calculator has been fully integrated and calculator screens are given to introduce computations. Everything the technical student may need is included, with the emphasis always on clarity and practical applications.
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About the Author
Paul A. Calter is a Visiting Scholar at Dartmouth College and Professor Emeritus of Mathematics at Vermont Technical College. He is a book review editor of the Nexus Network Journal and has interests in both the fields of mathematics and art. He received his B.S. from Cooper Union and his M.S. from Columbia University, both in engineering, and his Masters of Fine Arts Degree from Norwich University. Calter has taught mathematics for over twenty-five years and is the author of ten mathematics textbooks and a mystery novel. He has been an active painter and sculptor since 1968, has had many solo shows and participated in dozens of group art shows, and has permanent outdoor sculptures at a number of locations. Calter developed a course called "Geometry in Art & Architecture," which he has taught at Dartmouth College and Vermont Technical College, and he has taught at Dartmouth College and Vermont Technical College, and he has given workshops and lectures on the subject. Calter's own art is concerned with astronomical and geometric themes; he searches for a link between the organic and geometric basis of beauty, what has been called the philosopher's stone of aesthetics.
·This edition contains a new chapter on statistics as well as a new section on straight line graphing.
·Many of the exercises have been revised to ensure they are scaffolded in difficulty level from easy to challenging.
Clarity of presentation: This is the feature most mentioned by reviewers, and has obvious benefits to students and instructors.
Technical Applications: The technical applications provide motivation for the student, and examples for an instructor who may not have a technical background. Additionally, an Index to Applications aids in finding applications in a particular field, such as electrical technology.
Estimation: Shows a student whether an answer is reasonable or not reasonable. They show common pitfalls for both student and instructor and are flagged and boxed, wherever appropriate.
Formulas used in the text are boxed and numbered, and listed in the Appendix as the Summary of Facts and Formulas.
Writing, Projects, Internet: Every chapter concludes with a section of optional enrichment activities. Many students are attracted to the magic and history of mathematics and welcome a guided introduction into this world. These writing questions aim to test and expand a student's knowledge of the material and perhaps explore areas outside of those covered in class while team projects foster "collaborative learning." | 677.169 | 1 |
Tailored to both the specification and the tier, this Student Book delivers exactly what students and teachers need to cover the unit in exactly the right depth.
Synopsis:
* Supports teachers' understanding of AO2 and AO3 through clearly labelled AO2/3 questions in the exercises. * Packed with graded questions reflect the level of demand required, so students and teachers can see their progression. * Includes worked examples throughout the book break the maths down into easy chunks. * Uses feedback to highlight common errors | 677.169 | 1 |
Video instruction is a part of all curriculum in grades 7-12. However, in math and science, there's nothing like video and interactive multimedia, in conjunction with text, to help explain basic principles. The student has access to immediate assistance by our math teachers if further clarity is necessary.
Calvary Christian School emphasizes excellence in understanding principles of mathematics and in science at an early age.
Our goal is to aid the student in achieving proficiency in subjects which prepare them for college credit while attending CCS.
Algebra expands upon abstract concepts such as problem-solving and using variable expressions to represent unknown quantities. This subject is typically reserved for 9th-grade students. However, at CCS, students begin 8th-grade study of this vital elementary subject. With hands-on attention by our qualified mathematics teachers, this subject is not a problem to master for our young mathematicians. | 677.169 | 1 |
Format
508 products
This is the third book in the Life of Fred Pre-Algebra series. Join Fred Gauss, a child prodigy teaching at Kittens University, as he guides your students through functions, word problems, economic... Read More
Join Fred Gauss, child prodigy teaching at Kittens University, as he guides students through conversion factors, the coefficient of friction, square roots, and much more! Get the entire Life of Fred... Read More
Join Fred Gauss, a child genius who teaches at Kittens University, as he guides your children through irrational numbers, exponential equations, and much more! Get the entire College Prep series and... Read MoreThis book, along with Decimals and Percents, is the perfect preparation for algebra. Join Fred Gauss, a child genius who teaches at Kittens University, as he guides your students through common... Read More
This is the second book in the Life of Fred College Prep Math series. Join Fred Gauss, a child prodigy who teaches at Kittens University, as he guides your students through adding and subtracting... Read More
You can't miss out on this Awesome High School Digital Textbook Bundle! It provides you with 26 comprehensive Secondary Digital Texts and includes both the Student and Teacher Edition of each DigitalAn entire year of high school British history curriculum in an easy-to-teach and comprehensive volume. Students will complete this course knowing the rise of the British Empire that has influenced... Read More
Analytical Grammar teaches English grammar, punctuation, and usage. It is designed to be taught in three grammar "seasons" (see timeline) over three years; however, it is flexible enough to be... Read More
Four titles are combined for a full year of study. Building Blocks in Life Science delves deeper into biology and God's amazing design of life; The Genesis of Germs focuses on the spread of germs unique curriculum touches on four fascinating branches of science! With studies based on 4 of the engaging, full-color Wonder series books rather than tedious textbooks, this course adds a little... Read More
Higher Altitudes in Algebra I presents and builds upon the fundamentals of beginning Algebra to prepare the student for the more sophisticated algebraic mathematics encountered in Algebra II. 180... Read More
Friendly Chemistry is a truly unique approach to teaching introductory chemistry. Used by home schoolers and charter, public and private school students world-wide for over ten years, Friendly... Read More
Whether looking at stars or checking the local weather, these award-winning, full-color Wonders books show you the science within and beyond the sky! Learn about weather maps, how to build a simple... Read More
This is the same book used in elite private schools across the country. This best-selling book uses pop culture, intelligent trivia, and odd news events to make learning grammar fun. Over 2,000... Read More
Lapbook Journals are designed to bring a more hands on learning aspect to all levels of Apologia science. This product includes 944 pages of helpful and organized tools for improved retention of this... Read More
Apologia's Physical Science curriculum comes to life with this great tool. This product includes 588 pages of helpful and organized tools for improved retention of this challenging topic ~ you choose... Read More
Voice Lessons to Teach You How to Sing With Confidence Whether you are new to singing, or already sing and want to learn how to sing better, eMedia Singing Method can help. Professional voice teacher,... Read More | 677.169 | 1 |
In this book MathCAD work sheet will be used as a teaching and learning tool for the design and analysis of reinforced concrete beam using ACI code of design. The book has two sample problems that analyze and design both regular and irregular beams. First sample problems explain the use of MathCAD in design and analysis of simple rectangular beam. The 2nd sample problem explain the use of MathCAD in design and analysis of single and double reinforced regular beams such as rectangular beams and irregular beams such as T-beam, L-beam, I -beam and other shapes. In addition to bending reinforcement calculation, the work sheet will calculate the spacing requirement foe shear reinforcement and it will check beam adequacy for the shear. This MathCAD work sheet is intended for civil engineering students and professionals. Civil engineering students can use the work sheet to solve and understand the design and analysis of regular and irregular beams. This worksheet will be followed by other worksheets such as: Design of one way and two way slabs, design of axial and biaxial columns, and design of square, rectangular, and combined foundation.
"synopsis" may belong to another edition of this title.
About the Author:
Mohammed Bin Salem is currently an Associate Professor in the Civil Engineering department at the University of Qatar. In 1992, he received his Ph.D. in civil engineering from Catholic University of America. His research interests include earthquake,design of concrete and steel structures. MathCAD applications in analysis and design | 677.169 | 1 |
Curriculum Reform at UNL
If you attended UNL or taught mathematics at UNL more than five years ago, then you probably remember taking or giving tests that announced prominently, "NO CALCULATORS ALLOWED". Those who were part of the department more than fifteen years ago should have very little memory of computers in the department and no memory of computing being used in our undergraduate curriculum. All that has changed.
The past decade has witnessed significant changes in how we teach mathematics and technology issues has driven much of the change. In 1989, Professor Tom Shores received an NSF grant that led to the creation of the department's computer lab. Eventually, this lead to profound changes in instruction in differential equations (Math 221) and matrix theory (Math 314) and many upper division courses. Today, both of these courses have a lab fee and students are expected to work on extended writing projects which require the use of a computer algebra software such as Maple.
In 1993, the department began experiments with graphing calculators. Now all sections of our traditional calculus course (Math 106, 107, 208) use graphing calculators, assign writing projects, promote group work and use a "reform" textbook known popularly as the "Harvard calculus". More recently, John Orr has led the development of "Gateway Exams" that are given over the World Wide Web. The department's gateway exams are proficiency exams on finding derivatives and integrals. Students can repeat an exam daily for about four weeks but they must pass at a specified high level (about 80%) in order to pass the course. Several of our precalculus courses are also using graphing calculators and developing gateway exams.
Perhaps the most ambitious curriculum reform effort of all is being led by Steve Dunbar (and Bob Fuller in Physics). Multimedia Mathematics is a $1,000,000 grant from NSF that is joint with faculty at Oklahoma State University. One of only 7 grants out of 191 applicants, the project seeks to use tehnology as a means of adding mathematics to the curriculum in other disciplines and adding more science to our mathematics classes. | 677.169 | 1 |
Integrated algebra regents january 2012 answers explained
Additional worksheets aligned to the ccss are in progress. The banks include questions from Regents Exams dating back to 1866. . Common core state stan, dards classes, jMAP resources for the ccss include.KentChemistry home, chemistry Regents Exams with Explanation 2000 Regents MC Questions Explained 21 Complete Exams Explained, full natsume soseki kokoro pdf Regents Exams with Answers and Explanations (Multiple Choice and Short Answers).Resources may be downloaded using the links in the left column or below.Jmap offers teachers and other users of the Common Core State Standards free resources that simplify the integration of Regents exam questions into their curriculum.Science, history english, teachers, interdisciplinary jmap offers Regents exams in subjects other than math.Regents Exams in various formats, Regents Books sorting exam questions by ccss: Topic, Date, Type and at Random, Regents Worksheets sorting exam questions by ccss: Topic, Type and at Random,.Algebra I Study Guide, and, algebra I Lesson Plans. .You may also download ExamView software.Regents exam review classes, these resources are for students who have been taught all topics covered on the Regents exam, and are taking a Regents review/prep class.
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By using this website you agree to the use of cookies in accordance with the current browser settings.Download link is brokenThis mod may not be published on other sitesThis mod has been submitted with an invalid download linkThis edit was unallowed.Learn more about | 677.169 | 1 |
Developmental Courses
We offer three courses that will help prepare you for your
college-level courses in mathematics. These courses do not
count toward the mathematics credits required for graduation.
MAT 0028 Developmental Math 2
This course introduces
students to the basic concepts of algebra. Students
will simplify or perform operations on signed numbers,
radicals, polynomials, and expressions containing
exponents; factor polynomials; solve and graph linear
equations and inequalities in one variable; graph linear
equations in two variables; solve related applications.
MAT
0022C Developmental
Math Combined
Classroom-Based, Computer-Intense course that allows
students to work at their own pace. In this course, students
will learn operations on signed numbers, solving linear
equations and inequalities in one variable, operations on
polynomials, factoring, integer exponents, radicals,
graphing, and applications. AA degree-seeking students: Upon
successful completion of this modularized course ("S"),
student should register for MAT1033.
STUDY SESSIONS AND LABORATORY HOURS
All college preparatory mathematics courses have required
study sessions and laboratory hours. The study sessions are
designed to give students an opportunity to communicate with
each other about their course work, to get individualized help
from the study session facilitator, to review for exams, and
in some cases to work on course-related projects. The
laboratory hours are independent of the study sessions and may
be completed in the Mathematics Laboratory. The Mathematics
Laboratory is located in room 2223. Its hours of operation
are: MTWR 8:000 AM -9:00 PM, F 8:000 AM - 4:00 PM, SAT 8:00
AM-4:00 PM. You do not need an appointment. In the math lab we
have computer software, tutors, graphing calculators, and
videotapes to help you attain the goals of your mathematics
course. You will need to check in and out of the lab each time
you are there to earn the credit for the lab hour
requirement. Students taking MAT 0024 in eight
weeks must complete their lab hour requirements by the end of
those eight weeks.
MAT 0029
Developmental Math-to-Stat
Math courses for academic
pathways leading to statistics or liberal arts math. The purpose of the course
is to provide college-level (math) content stretched out across one
developmental course and one college-level course over a 16-week timeframe.
Students in this pathway are choosing majors that do not require MAT1033/MAC1105
or any course that need them as prerequisites.
MAT 0057 Developmental Math 3
Classroom-Based, Computer-Intense course that allows
students to work at their own pace. The student will learn
operations on signed numbers, solving linear equations and
inequalities in one variable, operations on polynomials,
factoring, integer exponents, radicals, graphing, and
applications. AA degree-seeking students: Upon successful
completion of this modularized course ("S"), student should
register for MAT1033.
College-Level Courses
MGF 1106 Mathematics for Liberal Arts I Topics include sets, logic, Euclidean geometry, probability, and
statistics. This course includes the competencies of the mathematics portion of
the CLAST (College Level Academic Skills Test) with the exception of algebra and
arithmetic competencies. (3 credits) Prerequisite: MAT 1033 or appropriate placement test score
MGF 1107 Mathematics for Liberal Arts II This course introduces the student to the concepts of
financial mathematics, linear and exponential growth, numbers and number
systems, history of mathematics, elementary number theory, voting techniques,
and graph theory.
Prerequisite: MAT 1033 or appropriate placement test score
MGF 1120 Basic Probability This course introduces the student to topics in probability and
statistics from a real-world perspective. (1 credit)
MAC 1105 College Algebra
This course introduces the student to the concept of functions
and their graphs. Students will graph linear, quadratic, rational, exponential,
logarithmic, radical, power, and absolute value functions and transformations;
perform operations on and compositions of functions; find the inverse of a
function; apply the laws of logarithms to simplify expressions and solve
equations; graph non-linear inequalities; solve related applications and
modeling problems. Prerequisite: MAT 1033 or appropriate placement test score
MAC 1105L College Algebra
Laboratory
This course is intended to accompany and support MAC 1105.
The competencies of this laboratory course have been introduced in the
accompanying lecture course. (1 credit/2
hours)
MTG 2204L Geometry for Educators Laboratory This is an accompanying laboratory to MTG 2204 in which students
perform constructions, work on projects and presentations, and use technology in
exploring geometric properties and patterns. (1 credit/2 hours)
MAC 2233 Business Calculus An introduction to the basic concepts of differential and
integral calculus for business majors. Topics include limits; continuity;
differentiation and integration of polynomial, logarithmic and exponential
functions with applications to business. (3 credits)
Prerequisite: MAC 1105 or equivalent or permission of the
department chairperson.
MAC 1140 Pre-Calculus Algebra
This course is primarily designed for students who are
majoring in areas that require one or more courses in the calculus sequence. The
student will analyze and graph algebraic, exponential, logarithmic,
piecewise-defined functions and conic sections. The student will solve
polynomial, exponential and logarithmic equations, as well as systems of linear
and nonlinear equations. The student will identify arithmetic and geometric
sequences and series and solve related problems. The student will use the
Binomial Theorem to expand polynomials and solve related problems. The student
will use mathematical induction to prove statements regarding the properties of
natural numbers. The student will solve applications and modeling problems
related to the above topics. (3 hrs. lecture) Prerequisite: MAC 1105 or equivalent or permission of the
department chairperson.
MAC 1147 Pre-Calculus Algebra and Trigonometry Topics include all of the topics of MAC 1114 and MAC 1140 (5
credits)
Prerequisite: MAC 1105 or equivalent or permission of the
department chairperson.
MAC 2311 Calculus and Analytic Geometry I Introduction to analytic geometry; limits; continuity;
differentiation of algebraic and trigonometric functions; differentials;
introduction to integration and the Fundamental Theorem of Calculus;
applications of the definite integrals and derivatives. 5 Credits.
Prerequisites: MAC 1114 and MAC 1140, or MAC 1147, with a grade
of "C" or better or departmental permission. (5 credits)
MAC 2930 - Integrated
Precalculus and Calculus, Part 1 This is the first of a two course
sequence in which the student will explore topics from Calculus and Precalculus
in an approach that integrates the two. (5 credits)
Prerequisite: MAC1105 with a grade of C or
better or equivalent
MAC 2931 - Integrated
Precalculus and Calculus, Part 2 This is the second of a two course sequence in
which the student will explore topics from Calculus and Precalculus in an
approach that integrates the two. (5 credits)
Prerequisite: Integrated Precalculus and
Calculus, Part 1, with a grade of C or better. | 677.169 | 1 |
standards, curriculum, and teaching
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Draft progressions on high school Algebra and Functions
I'm pleased to be able to give you the draft progressions on Algebra and Functions. These progressions are somewhat different from the K–8 progressions. Since the high school standards are not arranged into courses, the progressions are really more like descriptions than progressions; they are not in any particular curricular order. Furthermore, because each one covers a topic that occupies a large part of the high school curriculum, it gives less detail about how each standard might be addressed or how different standards might be arranged into various different curricular implementations.
13 thoughts on "Draft progressions on high school Algebra and Functions"
We (a college where I work as adjunct) are planning a three credit course for those math folks who would like to become "Math specialists" in school districts. This would include math coaches, math administrators, math coordinators and all the various names describing these leaders.
We are looking for a suitable book for the course. The problem I am having is the currency of these type books. We would like a CCSS perspective hence a Copyright after 2011. Any suggestions?
My name is Matt Friedman and I am an editor in Scholastic's Classroom Magazines' division. I am always curious to hear what resources teachers find helpful, and I wondered if you gotten to look at the book Bill suggested below. Could you you know if you found the book to be useful or if you found other resources that were out all in planning the course you discussed above?
Also, what university do you work for? The course you mentioned sounds very useful— I'm interested to learn more about it. Any information you can provide would be appreciated.
I assume you are talking about elementary grades here? If so, you might want to look at Sybilla Beckmann's book, since she was involved in the writing of both Curriculum Focal Points and the Common Core, and there is probably some harmony.
One more thing… page 12 of the Algebra Progressions says "Give Example in Margin". Do you have any good ideas for an example?
(Maybe I should have waited and put everything in one message. There are a couple of other small typos, let me know if you want them before the next draft.)
On page 10 of the Algebra document, there is an arrow connecting x^2=4 to x=+-2. Are we considering this standard notation that a student should know and use, or is this considered more short-hand for mathematicians that already know the math? if we are considering this standard notation, should we start using it in earlier grades – some places we already see elementary teachers doing this, but then some teachers incorrectly drop the arrow to just an equals sign.
What are your thoughts? | 677.169 | 1 |
In certain courses, all it will take to move an exam is take note using, memorization, and recall. Having said that, exceeding in a math class will take a distinct kind of energy. You cannot simply just show up for a lecture and enjoy your teacher "talk" about calculus and . You learn it by accomplishing: paying attention in school, actively researching, and resolving math issues – even though your teacher has not assigned you any. For those who find yourself battling to perform properly with your math course, then pay a visit to very best web site for resolving math troubles to understand how you can become a much better math university student.
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Math funny right triangle trigonometry problems programs adhere to a natural development – every one builds upon the awareness you have obtained and mastered within the previous system. In case you are getting it rough to comply with new ideas in school, pull out your aged math notes and evaluation earlier material to refresh your self. Be sure that you satisfy the prerequisites just before signing up for just a class.
Evaluate Notes The Evening Just before Class
Dislike any time a instructor phone calls on you and you have forgotten how you can address a specific issue? Steer clear of this minute by reviewing your math notes. This can assist you identify which concepts or inquiries you'd choose to go over in school the following day.
The thought of executing homework each individual night could seem irritating, but when you need to reach , it is actually important that you continuously observe and learn the problem-solving strategies. Make use of your textbook or on the web guides to operate through prime math challenges over a weekly foundation – even when you've no homework assigned.
Use the Nutritional supplements That include Your Textbook
Textbook publishers have enriched modern-day publications with extra substance (which include CD-ROMs or on line modules) which can be accustomed to help college students acquire more follow in . Some products might also incorporate a solution or explanation guide, which may enable you to with functioning by way of math troubles by yourself.
Examine Ahead To remain Ahead
If you prefer to reduce your in-class workload or the time you invest on research, make use of your free time after school or around the weekends to read in advance towards the chapters and concepts that could be covered the next time you will be at school.
Assessment Old Exams and Classroom Illustrations
The do the job you need to do at school, for homework, and on quizzes can supply clues to what your midterm or closing examination will glance like. Make use of your outdated tests and classwork to produce a private review guidebook for your impending exam. Appear in the way your trainer frames concerns – this is often probably how they are going to look with your take a look at.
Learn to Function From the Clock
This is a well-known analyze idea for men and women getting timed examinations; in particular standardized checks. If you have only 40 minutes for any 100-point check, then you can certainly optimally expend 4 minutes on each and every 10-point problem. Get data regarding how very long the exam are going to be and which varieties of issues will probably be on it. Then strategy to assault the better inquiries to start with, leaving on your own enough time for you to expend within the more tough ones.
Improve your Resources to acquire math homework enable
If you are getting a hard time comprehending ideas in class, then be sure you get assistance beyond course. Ask your buddies to produce a review group and pay a visit to your instructor's office environment several hours to go in excess of tricky troubles one-on-one. Go to examine and overview periods whenever your teacher announces them, or employ the service of a personal tutor if you need 1.
Talk To By yourself
Any time you are examining challenges for an exam, consider to elucidate out loud what system and techniques you utilized to get the answers. These verbal declarations will appear in handy all through a examination once you must remember the measures you'll want to take to find a solution. Get further practice by making an attempt this tactic which has a close friend.
Use Analyze Guides For Additional Exercise
Are your textbook or class notes not assisting you fully grasp whatever you ought to be finding out at school? Use research guides for standardized exams, such as the ACT, SAT, or DSST, to brush up on old content, or . Review guides ordinarily arrive geared up with comprehensive explanations of the best way to solve a sample challenge, , and you also can often locate where is the greater obtain mathproblems. | 677.169 | 1 |
Student and educator editions of pre-calculus textbooks bring real world math into an engaged classroom. Many texts integrate relatable examples and applications into everyday classroom activities. Houghton Mifflin Harcourt is a specialist in delivering quality core curriculum content, as well as enhancement materials for the digital age.
A robust and positive pre-calculus experience will serve as a sturdy foundation for the calculus course to come. The preparatory calculus curriculum is part of the path to STEM studies or challenging college mathematics. That's why we offer pre-calculus texts containing succinct explanations, extensive problem-solving examples, and constructive practice to facilitate student comprehension and accomplishment. Pre-calculus educator resources are well-aligned with student editions, often providing tips for each stage of critical instruction. Manuals can also assist educators in uniting math instruction and technology into a complete system to guide pre-cal student success.
Educator Resources for the Pre-Calculus Curriculum
Through digital tools, a math classroom can be transformed into an interactive environment. We provide educator resources for the pre-calculus curriculum in a range of valuable mediums. Whether the class objectives require an app, an eText, or a print title, our tools are designed to work in sync with educator goals. Students can benefit from a manual that demonstrates graphing calculator procedures. Innovations in learning technology will also help students absorb material – from linear to trigonometric functions. We invite you to shop by grade or subject to find the algebra resources, geometry textbooks or pre-calculus content you require. Please contact us if you have any questions about our educator resources and textbooks for teaching pre-calculus. We're here to support your specific educational mission | 677.169 | 1 |
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
Bob Miller's basic math and pre-algebra for the clueless
"If you suffer from math anxiety, then sign up for private tutoring with Bob Miller! Do mathematics and algebraic formulas leave your head spinning? If so, you are like hundreds of thousands of other students who face math-especially, algebra-with 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 knowledge, empathy, and fun to math and pre-algebra. He breaks down the learning process in an easy, non-technical way and builds it up again using his own unique methods"--Publisher website (February 2008).Read more...
Abstract:
Algebra and calculus are tough on high school students. Written by an author with more than 30 years' teaching experience, this book breaks the tougher areas of mathematics so you can master the materials and ace each test.Read more... | 677.169 | 1 |
Description of the book "The GED Math Problem Solver":
The GED Math Problem Solver integrates problem-solving and reasoning strategies with mathematical skills using problems encountered in everyday life. This text builds understanding of mathematical relationships by focusing on problem-solving skills, developing estimation and mental math strategies, and integrating algebra, geometry, and data analysis with arithmetic.FEATURES
25 lessons combining instruction, practice, and reviewComplete answer key, including solutionsCumulative review and GED practice at the end of each lessonTest-taking lessons and practiceExercises using data and graphs collected in the appendixCalulator exploration using the Casio fx-260Full-length GED Mathematics practice test
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McGraw-Hill
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by 瘋數學 Provided under first-year junior semester math exercises with answers and explanations.
Is a national middle school after-school review, the best practice exam assessment tool.
Total of multiple choice, fill in the blank with the word problems three kinds of questions. Save chapter contains all 900 questions for users to practice.
After each chapter is selected, will automatically generate 5 questions, each question practice. If you want to control solution to a problem, you can press the button to display the contents of problem-solving | 677.169 | 1 |
The book begins at the level of an undergraduate student assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, Lebesgue integral, vector calculus and differential equations. After having built on a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis, as understood by a mathematician today.
Igor Kriz studied in Prague, Czech Republic. He has lived in the United States since 1988 and been teaching at the University of Michigan since 1994. His main interests are geometry and algebraic topology. Aleš Pultr studied in Prague and has been teaching at the Charles University since 1961. His main interests are point-free topology, category theory and combinatorics. Both authors have taught numerous courses of mathematical analysis.
"The book is intended as a second-year course of mathematical analysis for advanced undergraduate students. … The volume is addressed to undergraduate students seriously interested in mathematics and is accessible to students before they start taking graduate classes. Researchers in pure and applied nonlinear analysis will find interesting material in this volume." (Teodora-Liliana Rădulescu, zbMATH, Vol. 1279, 2014)
"The authors included in their book some topics from topology, calculus of real functions of one and several real variables … elements of functional analysis, as well as some applications. … the present well written book is a valuable addition to the existing ones on similar topics. It can be used by graduate students in mathematics and researchers in mathematics and other areas … . The instructors can recommend the book as a supplementary material for their courses." (S. Cobzaş, Studia Universitatis Babes-Bolyai, Math, Vol. 58 (4), 2013) | 677.169 | 1 |
I'm getting really bored in my math class. It's permutation math gmat , but we're covering higher grade material. The topics are really complicated and that's why I usually sleep in the class. I like the subject and don't want to drop it, but I have a real problem understanding it. Can someone guide me?
Hey brother. Let me tell you one thing, even experts in this field sometimes lag in a particular topic. Mathematics is such a vast subject, that it sometimes becomes impossible to excel every part with equal ease. If you are facing problems with permutation math gmat , why don't you try Algebra Helper . This program has rescued many colleagues of mine and I have used it a couple of times as well. I was quiet satisfied with it.
Algebra Helper is the program that I have used through several math classes - Pre Algebra, Intermediate algebra and College Algebra. It is a really a great piece of math software. I remember of going through problems with factoring polynomials, binomial formula and radical expressions | 677.169 | 1 |
Math 3C
Fall 2011 Course Syllabus
Course: Math 3C is the preparatory course for Math 10A (Calculus). The goal of the course is to develop the mathematical knowledge and skills, as well as the analytical and problem-solving skills,
necessary for the study of calculus. Unlike some courses, attendance is required for all lectures, 4th
hours, and discussion sections. The 4th hour is used as an
additional lecture, since the course syllabus cannot reasonably be covered
using only the scheduled lecture hours. Discussion sections are used for quizzes.
Calculators: Graphing calculators are recommended (but not required) for use in class and for homework. The calculator should be used as an aid in learning concepts, not just as a means of computation. Help with using TI graphing calculators will be available in the
Calculus Tutoring Lab.
Note: The use of calculators will not be permitted during the midterms or the final exam.
Homework: Tentative homework assignments and their due dates have been posted on the course homework page. Any changes to the assignments will be announced in class and in the announcements section on the main page of the course web site. Late assignments will not be accepted; however, the lowest homework score will be dropped. Selected problems on the assignment will be graded. While your work will be graded for correctness, keep in mind that presentation and organization directly affect your grader. To assist your grader, you should keep all problems in the same order as the assignment list and ordered vertically on your page. If a problem is omitted, it should still appear in the correct order. All work must be on full-sized notebook paper and all pages must be stapled together. Write your name (last name first), your Discussion Section (A01, A02, etc.), and the homework assignment number in the upper right corner. Homework will be returned in section the following week. Watching someone else do a problem or "understanding" the solution in a Solutions Manual does not indicate mastery—you must be able to begin a solution on your own and carry it to completion without help before you can claim mastery of that problem. Ask questions during office hours about homework problems that have caused you difficulty and indicate what attempts you have made to solve the problems. A thorough understanding of the homework problems and their solutions will prepare you well for the midterm exams and the final exam.
Quizzes: There will be seven discussion quizzes. Discussion quizzes are group work quizzes, where groups of 3 to 4 students work on a set of assigned problems and turn in one written set of solutions as a group during section. There will be no makeup discussion quizzes, however, the lowest discussion quiz grade will be dropped. The TA and student interns will be available to facilitate these assignments and assist the groups during the discussion. See the course calendar for the dates of the discussion quizzes.
Midterm Exams: There will be two midterm exams during the quarter. Note that the midterms are scheduled during the fourth hour lectures; see the
course calendar for the dates.  There will be no makeup exams.
Final Exam: The final examination will be held at the following date and time.
It is your responsibility to ensure that you do not have a schedule conflict involving the final examination; you should not enroll in this class if you cannot
sit for the final examination at its scheduled time.
Grading: Your course grade will be determined by your cumulative average at the end of
the term; your cumulative average will be the best of the following two weighted averages.
Note: Since there are no makeup exams, if you miss an exam for any reason, then your course grade
will be computed with the final exam counting 55% of your weighted average.
Academic Dishonesty: Academic dishonesty is considered a serious offense at
UCSD. Students caught cheating will face an administrative sanction which may include suspension
or expulsion from the university. (Click here for more information.) | 677.169 | 1 |
Trigonometry
Notes, Problems, and Exercises
"It has the content, examples, and exercises to supplement a standard text or lectures. The greater value is on subjects not appearing in many introductory texts such as Ptolemy's Theorem, Napoleon Circles, and pedal triangles."
MAA Reviews
This book provides a thorough, intermediate-level yet concise course in Trigonometry for use in colleges. There are 37 short chapters, each treating one specific theme and containing worked examples and easy exercises. Central to the work are the trigonometric properties of triangle ABC and its associated points. A small appendix contains some Spherical Trigonometry with interesting problems related to the earth; a larger one for enthusiastic students provides further lengthier exercises for extra practice, and full solutions are supplied in the conclusion.
Compared with other books on Trigonometry, this book covers the vast spread of topics. Especially, the author reminds readers of the historical importance of theorems enunciated by such contributors as Ptolemy, Euler, Morley, etc. Their names not only invite the readers to appreciate the beauty of these results, but also direct readers to mystery unknown. | 677.169 | 1 |
Teaching Assistants
Course description
This is the basic foundational class of the Mathematics in Finance program for the sell side of the
quantitative finance industry. It covers the basics of forwards and options, and how they are used in the
modern finance industry. Mathematical tools are developed as needed, including elements of partial
differential equations and stochastic calculus. Students will do some light computing using Microsoft
Excel and VBA and a little C++. See the detailed syllabus for more information.
Prerequisites
A mastery of multivariate calculus, multivariate probability, and linear algebra.
Basic computing skills including some programming experience in some language such as VBA,
Matlab, C/C++, Java, Python, etc. | 677.169 | 1 |
Math2: Monday 6-20-11
1.
Monday June 20, 2011 Math 2 Objectives and Agenda
2.
Math 2 Course Objectives Number sense: Statistics and probability Represent and use numbers in equivalent Collect, organize and represent data forms Read and interpret data representations Understand meanings of operations and Describe data using numerical descriptions, how they relate to one another statistics and trend terminology Compute fluently and make reasonable Make and evaluate arguments or statements by estimates applying knowledge of data analysis Know and apply basic probability concepts Patterns, functions and algebra Geometry and measurements Explore, identify, analyze, and extend Use and apply geometric properties and patterns in mathematical and adult relationships to describe the physical world and contextual situations identify and analyze the characteristics of Articulate and represent number and data geometric figures relationships using words, tables, and Use transformations and symmetry to analyze graphs mathematical situations Recognize and use algebraic symbols to Specify locations and describe spatial model mathematical and contextual relationships using coordinate geometry and situations other representational systems Analyze change in various contexts Understand measurable attributes of objects and the units, systems, and processes of measurement and apply appropriate techniques, tools and formulas to determine measurements
3.
What are we doing today? Today's Objectives We will get to know our classmates. We will graph and interpret the things we have in common and what makes us unique. We will set goals for this class and our education in general. We will check our current understanding of the math concepts covered in Math 2.
4.
Welcome to Math 2!Here are some things you should know… Class schedule Materials to bring with you What will we be studying? Class expectations Class website and Andrea's contact info: | 677.169 | 1 |
Learning Objective
An important goal of the Discrete Mathematics is to develop student's ability to think abstractly, this requires that students learn to use logically valid forms of argument. Students will get fundamental concepts about sets, relations, functions, and mathematical induction and they will use their applications in Computer Science. | 677.169 | 1 |
Download A Student's Guide to the Study, Practice, and Tools of by Donald Bindner PDF
A Student's advisor to the examine, perform, and instruments of recent arithmetic presents an obtainable creation to the realm of arithmetic. It bargains easy methods to examine and write arithmetic in addition to the right way to use a number of mathematical instruments, from LaTeX and Beamer to Mathematica® and Maple™ to MATLAB® and R. in addition to a colour insert, the textual content contains routines and demanding situations to stimulate creativity and increase challenge fixing talents. the 1st element of the publication covers concerns touching on learning arithmetic. The authors clarify tips on how to write mathematical proofs and papers, the way to practice mathematical examine, and the way to offer mathematical displays. the second one part specializes in using mathematical instruments for mathematical typesetting, producing info, discovering styles, and lots more and plenty extra. The textual content describes how one can compose a LaTeX dossier, supply a presentation utilizing Beamer, create mathematical diagrams, use computing device algebra structures, and reveal rules on an internet web page. The authors conceal either well known advertisement software program courses and loose and open resource software program, reminiscent of Linux and R. displaying the right way to use know-how to appreciate arithmetic, this consultant helps scholars on their approach to changing into specialist mathematicians. For starting arithmetic scholars, it is helping them examine for assessments and write papers. As time progresses, the publication aids them in acting complicated actions, similar to machine programming, typesetting, and study.
During this consultant, you will how one can use CEDA, the popular approach for dossier entry throughout a community, in addition to how one can use the conventional approach to relocating a SAS dossier throughout working environments.
This ebook is designed to aid clients speedy study the first positive aspects of the PSS program, a point-and-click interface for strength research and pattern dimension selection. The software program presents computations for numerous statistical analyses, together with ttests; self assurance durations and equivalence assessments for ability; designated and approximate assessments of proportions; a number of regression and correlation; one-way research of variance; linear types; and rank assessments for evaluating survival curves.
This e-book represents an integration of concept, equipment facts. it's written for researchers and practitioners within the finance undefined, educational researchers in economics and finance, and complex MBA and graduate scholars in economics and finance.
If you have a simple realizing of facts and wish to benefit SPSS on their lonesome, utilizing IBM SPSS facts, moment variation via interpreting their very own information, instead of easily commencing latest databases.
Additional info for A Student's Guide to the Study, Practice, and Tools of Modern Mathematics
Sample text
What was the K¨ onigsberg Bridge Problem? Who solved the problem? What branch of mathematics did the solution inaugurate? 7. Who said, "It is impossible to be a mathematician without being a poet in soul"? 8. Who was Paul Erd˝ os? What branches of mathematics did he create? Explain what an Erd˝ os number of a mathematician is. 9. Who showed how to construct a regular heptadecagon (seventeen sides) with straightedge and compass? How old was this mathematician at the time? 10. Who said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"?
Contributing questions to the problems sections of journals, and solving such problems, are also important. Typically, a faculty member must produce some publications in order to be considered for tenure (a career-long position) at an academic institution. Other achievements typically necessary for tenure are high-quality teaching and service to the department and the institution. Research is often a collaborative effort. For example, the classification of finite simple groups, completed in 1986, was the joint effort of over 100 mathematicians from about ten nations, working over 40 years, writing about 10,000 pages of journal articles.
Being a mathematics professor sounds like a lot of work, doesn't it? It is. But the work is both rewarding and important. One rewarding aspect of being a university professor is teaching and supervising enthusiastic and talented young people, some of whom will become professional mathematicians. One important aspect of being a professor is holding a high standard in terms of academic integrity. As professors, we want our ideals of honesty, responsibility, humility, and curiosity to take hold in our students. | 677.169 | 1 |
Lines, the Straight Kind. Quadratic Equations. Inequalities, Linear and Quadratic. Absolute Value. Exponents, Negative and Fractional. Geometric Formulas and Facts You Must Know. Distance Formula, Midpoint, Circle, and Parabolas. Functions. Linear Systems and More About y=x2. Trigonometry. Curve Sketching. Modern Logarithms. Parabolas II, Ellipses, and Hyperbolas. Writing Functions of x, the Algebraic Part of Calculus Word Problems. SAT Algebra: It Will Also Help Those of You Are Past the SAT. The Binomial Theorem. | 677.169 | 1 |
This directed case study is designed for an intermediate algebra course, appropriate for high school mathematics and lower division college mathematics. The ideal class size for running the activity is generally between 16 and 20 students. The main objective is to help students understand the concept of the slope of a line and be able to explain the definition in an applied scenario. After completing this case study, students should be able to graph a line and linear inequality. | 677.169 | 1 |
Each team will try to identify a series of mathematical functions
given either a table of values, the graph, or both.
Participants
Teams of up to six.
Procedure
Each team will have from 4 to 6 members
with at least one student from each of grades 11, 12 and OAC. The
remaining team members may be from any grade level.
Each team will be seated at a table, and given an envelope with
15 pages inside. On each page will be an accurate graph of a mathematical
function; and, where necessary, a table of values for the unknown
function. The functions will be in one of six categories: linear,
quadratic, polynomial, rational, trigonometric, exponential and
logarithmic. In some cases, the category will be given to you. The
object of the competition is to identify the equations of the functions
within the time limit of 30 minutes.
The team can divide up the work as they see fit and discuss, cross-check,
argue and so on as they come to their collective conclusions about
the unknown functions.
Training
for the Event
A computer program called Identify for PC's running under
DOS is available for download. It will allow you to practise your
skills in identifying functions from each of the six categories.
This program presents the unknown function either as a graph or
as a set of numerical values. You can also ask for the values of
any point on the graph or trace the curve and get a running tabulation
of points on the curve. This information then allows you to establish
the equation. You can input the equation when ready and the computer
will tell you if it is correct or not. The authors of the program
have placed it in the public domain, so that you may make as many
copies as you need.
Number of Functions in the Program:
Linear 40
Linear & Quadratic 60
Polynomials 84
Rational 32
Trigonometric 88
Exponentials and Logarithmic 24
A set of bonus questions will be taken directly from some of the
more challenging functions in the program.
This public domain program will seem archaic to students who have
known only Windows. If anyone designs a more modern substitute and
is willing to share it, please contact
the web master.
Judging
Each question will be scored out of 5,
so that the maximum team score is 75 marks. This score will contribute
33.3% to the overall score for the Mathematics Triathlon. The other
components of the Triathlon are Lightening Calculators and
Serial Series - Serious Stuff.
Source
London District Science Olympics. This
event was created by Eric Wood. | 677.169 | 1 |
Sharing thoughts on the teaching of mathematics
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A Hodgepodge of Ideas?
When I started teaching Precalculus, I thought it was no wonder the students were confused. It felt as if we jumped from topic to topic – complex numbers, trigonometry, logarithms, polar coordinates, etc. – and spent only a few days on some of them. It was just last year that I was finally able to really help my students see the connections.
For most of the course, we are studying the functions. My students will need to know them all pretty well for calculus. And the functions share many characteristics: they have a domain and a range and a graph and can be shifted right or left, up or down, or reflected in the x– or the y-axis (although a reflection in the y-axis may not return another function.) The notation for these shifts and reflections is exactly the same, no matter what the function:
Students need so see how these are used with a number of different functions before they get they idea.
Those functions whose reflections in the line y = x is also a function have an inverse and the inverse works the same for all these functions: linear, square root, cubic, exponential, etc. When you reflect in this line, the y's become the x's and the x's become the y's. As a result:
We have an algorithm for finding the equation of an inverse that calls for swapping the x and the y and solving for the new y.
The dimensions of the domain of the parent function become the dimensions of the range of the inverse function. The dimensions of the range of the parent function become the dimensions of the domain of the inverse function.
The y's become the x's and the x's become the y's. The range becomes the domain and the domain becomes the range. The second bullet is a direct consequence of the first. Inverse functions are so useful that we truncate the trigonometric functions in order to create them, and we actually have a totally artificial, man-made inverse of the exponential function, namely the logarithmic function.
Some functions have a y-intercept, an x-intercept, vertical or horizontal asymptotes, or periods and amplitudes. Some are algebraic and some are transcendental. Each is needed to model some relationship in the real world: periodic functions for sound waves or business cycles, exponential (or logistic) functions for population growth, etc.
In A Tour of the Calculus, David Berlinski presents and discusses the Mandela of the Functions. I share the image from this book with my students. Here is a link to a web site where you can see how the functions come full circle. Page down once to see the diagram.
All of these are based on our Cartesian coordinate system, but there are functions whose equation are much simpler if expressed in terms of distance from the origin (pole) and direction from the positive x-axis (initial ray). Polar coordinates and the trigonometric form of a complex number have a lot in common, namely the Pythagorean Theorem. The work we have done on building the unit circle from the 45-45-90 and the 30-60-90 triangles sets the stage. This year, for the first time, my students knew the special right triangles by heart and so were easily to solve both of these types of problems.
As I wrote this I wondered why it was that I finally made these connections myself. Was it "just" another year of experience and reading books & blogs? I think I owe much credit to a summer course I took at a local college. For two weeks, a different very experienced teacher came in each day and shared their best lessons and ideas. It was an amazing experience and I still refer back to those notes frequently. Thank you, Art!
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3 thoughts on "A Hodgepodge of Ideas?"
Assuming this post was in response to the new blogger prompt, this is similar to the post I considered writing in response to 'pre-calculus or algebra 2 is random topics.' Sets and functions can make even the random stuff seem not so random – for example, using the notation P(event) to emphasize probability as a function | 677.169 | 1 |
MTH 208 Free DQ Guide
Part 1: Within our library, on the main screen under the title Center for Mathematics Excellence, access 'Building Math Confidence', scroll down, under the category of 'Overcome Math Anxiety' access 'learn more'. Then, take the 'Anxiety Quiz'. Discuss your findings
Part 2: Imagine your younger relative—of middle school age—was taking an algebra course and asked for your help. How would you teach the multiplication of polynomials to him/her?
Part 1: Experience a Live Math Tutoring session by clicking the Live Math Tutoring link (under Useful Links on the Materials tab). Ask the tutor to assist you with a Week One problem or concept. Post the problem or concept a tutor helped you with, the date and time of the tutoring session, and how valuable you think having live tutors with 24/7 availability will be for your success in this course. Please also share which whiteboard tools you found to be most helpful in your discussions with the tutor.
Part 2: Do you always use the property of distribution when multiplying monomials and polynomials? Explain why or why not. In what situations would distribution become important?
Why is it important to follow the order of operations? What are some possible outcomes when the order of operations is ignored? If you invented a new notation where the order of operations was made clear, what would you do to make it clear?
What are the four steps for solving an equation? Should any other factors be accounted for when solving an equation? Should any factors be accounted for when explaining how to solve an equation? Explain your answer.
What resources are available to help you do well in this course? Which resources do you think will help you the most? Why? How do you plan to use the resources available to optimize your learning over the next four weeks?
What are the four steps for solving a problem? Should any other factors be accounted for when solving a problem? Should any factors be accounted for when explaining how to solve a problem? Explain your answer.
Imagine that a line on a Cartesian graph is approximately the distance y in feet a person walks in x hours. What does the slope of this line represent? How is this graph useful? Provide another example for your colleagues to explain.
If a line has no y-intercept, what can you say about the line? What if a line has no x-intercept? Think of a real-life situation where a graph would have no x- or y-intercept. Will what you say about the line always be true in that situation? | 677.169 | 1 |
Calculus with Analytic Geometry I Advice
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I would definitely recommend this course. Professor Capuano is one of the best math teachers I have ever had. She teaches like an old school math teacher, who first shows you the background for the methods and theorems shes teaching then shows you the process step by step, then she goes through example problems providing a multitude of different situations that you could come across on homework, quizzes, and tests. Her tests are very straight forward so you know what to expect unlike some teachers that try to confuse you by asking you a question in a totally out of the box way; before every test she will go over review questions the day be for the test. Overall she is a great teacher who makes taking the class worthwhile.
Course highlights:
Throughout this course I learned the proper methods to do anything from how to find derivatives and their inverse functions, also know as antiderivatives or integrals to finding the area under a curves by creating integrals and solving them. I also learned how important it is to focus on the method and/or the theorems behind the calculus and not just the answer; if you can't use the methods behind the calculus the end result really doesn't matter even if it is right.
Hours per week:
6-8 hours
Advice for students:
If you want to succeed in this course you need to keep pace with the professor as she teaches you one section you should do the homework for that section before the next class so that you are ready to move on when the teacher does. Also familiarize yourself with the programs that you will use during you lab class, such as Maple and Converge. Make sure you go to every lab class and turn in you lab every week; make sure you lab is formatted in the correct way. | 677.169 | 1 |
1)Familiarize students with the most popular numerical techniques for the
solution of mathematical problems.
2)Provide students with practical CE applications of the numerical
techniques that are taughtin the
course.
3)Familiarize students with the software Mathematica, which is a very
powerful tool for the solution of a wide range of typical mathematical
engineering problems.
Course
Rationale:
The
mathematical tools of this course will be very useful for solving many of the CE
problems that students will encounter in their courses, as well as after
graduation.In addition, the
software package will be of tremendous help in getting quick solutions to
complicated problems.
COURSE
OUTLINE
Weeks #
Topic
Chapter
# (Selected Sections)
1
Introduction
to Computing
·Ch. 1 - Mathematical Modeling & Engg.
Problem - Solving
·Ch. 2 - Computers and Software
2
Approximation
& Errors
·Ch. 3 - Approximation and Round-Off
Errors
·Ch. 4 - Truncation Errors and the Taylor
Series
3 & 4
Roots of
Equations
& CE
Applications
·Ch. 5 - Bracketing Methods
·Ch. 6 - Open Methods
5, 6 &
7
Linear
Algebraic Equation
·Ch. 9- Gauss
Elimination
·Ch. 10 - LU Decomposition and Matrix
Inversion
·Ch. 11 - Special Matrices and Gauss-
Seidel
8 & 9
Introduction
to Optimization
& CE
Applications
·Ch. 13 - One-dimensional Unconstrained
Optimization
10 & 11
Curve
Fitting
·Ch. 17 - Least-Squares Regression
·Ch. 18 - Interpolation
12 & 13
Numerical
Integration
& CE
Applications
·Ch. 21 - Newton-Cotes Integration of
Equations
14 & 15
Ordinary
Differential Equations
·Ch. 25 - Runge-Kutta Methods
·Course Grade Breakdown:
Attendance & Class/Lab participation:10%
Homework & Lab Assignments:15%
Quizzes:5%
Exam # 1:20%
Exam # 2:20%
Lab Final Test:5%
Course Final Exam:25%
·The University policy regarding excessive absences will be adhered to
strictly, which may result in (DN) grade. For their own benefit, students
are strongly encouraged to not miss any class or lab session. | 677.169 | 1 |
The Art of Mastering Resources
Algebra is a branch of mathematics. Algebras deal with variable activities in our day to day living. Algebra comes directly after learning the mathematics arithmetic.
Variable activities.
A variable is something that changes often. For instance trends going up or down, west or east, left or right. The weight of a person is a good example of a variable because it does not remain the same, it keeps on changing by getting lower or higher every time. Another example is the sun, it does not stay at the same place all day, it changes position from east to west due to the motion of Earth around the sun. Every time there is a change in the stock market from high to low. The amount of a worker's salary is an active variable because some salary changes according to the number of hours they work. The Ultimate Guide to Education
Algebra is also referred to as the study of activities that keep on changing with time. There are countless of applications of Algebra in our day to day lives. Study: My Understanding of Education
Elementary concepts needed to be taught before starting Algebra.
Elementary multiplication, addition subtraction and divisions.
Multiplication tables at least to 10.
Have the knowledge of drafting the entire factor of numbers, finding the greatest common factor (GFC )and the least common multiple (LCM).
Units.
Whole numbers.
Command of operations.
Multiples are added to many students in the fourth grade. Students should learn about multiples when they begin learning multiplications and get relaxed with it. The next thing after the students start learning multiples is knowing where and how to use multiples in mathematics. When the students get to be familiar with multiples, their core competencies of math are enhanced and can be able to predict times of two numbers properly without delaying.
Primary topics learned in Algebra.
Familiarity with variables.
Know the constants and the coefficients.
Writing Algebraic expressions.
basic linear calculations in one variable.
Balanced expression and Factorization.
Patterns like series and sequences in overall.
Why is Algebra hard?
Algebra is not hard. You can take it a challenging course. There are extremely general terminologies used in Algebra. Generic terms mean, for instance, terms or characteristics used to differentiate people from one another in a place where there are several similar characters with some similar characteristic. Some rules are to be followed in Algebra. If you follow these procedures, Algebra is not that tough.
In the elementary theory of numbers, finding the greatest common factor that divides two or more numbers without leaving a remainder is an important thing. During the fifth grade, students are introduced to factors finding, but some students in several schools are introduced to factor finding while in the late fourth grade. Having the knowledge about composite and prime numbers is the key to learning factoring of numbers. | 677.169 | 1 |
The aim of this book is to throw light on various facets of geometry through development of four geometrical themes.The first theme is about the ellipse, the shape of the shadow cast by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations.The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the Poincaré disc model of the hyperbolic plane is developed | 677.169 | 1 |
Matrix Applications
This blog contains matrix applications that I have found or developed to use in Linear Algebra or Mathematical Modeling. You'll need to know some linear algebra to understand these posts.
These are not in-depth explanations but just a taste to whet your appetite. There will be some errors. Gently point them out, and I'll fix them.
Saturday, June 3, 2017
Solving systems of equations
Nodal analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters. Nodal analysis creates linear equations using Kirchhoff's Laws of junctions and paths. This is a popular project for students who have studied some physics. One value of this project is the ability to create overdetermined, consistent systems of equations, which helps students understand rows of zeros in the RREF form of augmented matrices. This article on Nodal Analysis of Electric Circuits has a clear explanation.
Loop analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters. Loop analysis creates linear equations using Kirchhoff's Laws of loops. Again, a popular project for students who have studied some physics and also has the opportunity for overdetermined, consistent systems. Equivalent in results to nodal analysis, this could be combined or assigned separately. This article on Loop Analysis of Electric Circuits has a clear explanation.
Curve fitting - Using systems of equations a student finds the coefficients of a polynomial of degree n - 1 to fit n points. I don't think of this as a juicy application that gives the student an appreciation for how linear algebra is used in the world. Fitting an n-degree polynomial to m points using least squares or other methods is more likely to happen.
Thursday, June 1, 2017
Inner product spaces
Curve fitting using least squares - Uses matrix multiplication, inverses, and equations to find coefficients of a curve that fits a set of points. I have three misgivings about curve fitting with least squares: it can be done without any understanding, it is not representative of least squares problems in general, and the various spaces involved confuses the issue. Taking the last point first, the problem involves points in 2D or 3D space, matrices in n x k space where n is the number of points and k the terms of the curve being fit, and coefficients that live in k-space. If we are trying to fit a line, the coefficients are 2D, but that 2D space is not the 2D space of the original points.
If a student is assigned this project without have learned about projections, they can do the calculations anyway, since they just require matrix multiplication and solving systems of equations. The process of setting up the matrices does not promote deeper understanding of inner product spaces, and so if the student is going to fit curves, they might as well use Excel, which also doesn't deepen their understanding of linear algebra.
Finally, if a student learns least squares in this way, then they have difficulty transferring this concept to the solutions of noisy systems using least squares and don't think of least squares as a method of approximating solutions, but rather of fitting curves.
Friday, May 26, 2017
Matrix Operations
I've just finished teaching Linear Algebra twice since the beginning of the year, and I'll be teaching it again in the fall. It is time that I cleaned up my applications list and updated the project files. Since I am enumerating them, I might as well do it here. Here is part 1 on matrix operations. Others may be added later. Seriation in archaeology - Uses incident matrices, matrix multiplication and transposition, and the properties of symmetric matrices to determine relative ages of sites with common artifacts. I haven't used this in my classes yet, but there seem to be some good resources available including "Some problems and method in statistical archaeology," David Kendall, World Archaeology, 1969, which is available in JStor. It also appears in Gareth Williams Linear Algebra with Applications. Color manipulation in images - Uses matrix multiplication to alter colors in the RGB scale. This article by Paul Haeberli describes the 4x4 matrices needed to modify colors, including offsets. I haven't used this in class yet, but I could see this as a good project to have students work in Mathematica. It also is a companion for transformations in 3D graphics, which also use 4x4 matrices. The ability to use matrix multiplication to add vectors is common to both areas.
Image color conversion - Uses matrix multiplication to convert from, say, RGB to YIQ, color models. I haven't used this in class, but it shows up in Gareth Williams Linear Algebra with Applications. This article by Ford and Roberts describes a bevy of color models, and it seems that only some conversions are linear. Without a way to test whether the color conversion is correct, I don't see this as an interesting project. However, maybe Mathematica can render the other color models.
Transformations in 2D graphics - Uses matrix multiplication to apply rigid and non-rigid transformations to images. May or may not use projective coordinates, depending on whether translations are allowed. Resources abound.
Projection of 3D images onto the plane - Uses matrix multiplication to project 3D images onto the plane given the coordinates of the image and the location of the viewer. Uses projective coordinates. I use a paper written by Jeanie Mullen, one of my honors students. This project has worked best for students with programming backgrounds.
Two-port in an electrical circuit - Uses matrix multiplication to describe the change in voltage and current through a two-port or a series of two-ports. A simple application of Ohm's law that creates two linear equations that can be described using matrix multiplication. The equations relate the input current and voltage to the output current and voltage. This Wikipedia article has a table of many transmission matrices and their effect.
Wednesday, May 24, 2017
Eigenthings
Gould's accessibility index in a network - The process uses a modified adjacency matrix and the components of the eigenvector associated with the dominant eigenvalue. Students find this approachable and adaptable. Applications to historical geography, air traffic.
Discrete dynamical systems - Using linear algebra to study discrete dynamical systems comes in several flavors. Here are some projects that students find interesting and that differ from each other enough that they feel they are not repeating someone else's project.
Difference equations and the Fibonacci sequence - Using eigenvalues to write the product of the nth power of a diagonalizable matrix and an initial vector allows one to write a closed form for a recursive formula. Matrices of size 2x2 are needed to write the closed form of the nth Fibonacci number, but students can easily move from there to the closed form of 3rd and 4th order difference equations. This project is always chosen by some student even though it is not applied to a real-world situation.
Monday, May 22, 2017
Matrix Inverses
I separate these projects from those other using matrix operations, because I make a clear distinction between forward and inverse problems in my classes.
Cryptography - These projects come in two varieties: using modular arithmetic and not.
Matrices with |determinant| = 1 - Uses matrix multiplication to encode a matrix and multiplication of the inverse to decode. Any matrix with determinant 1 or -1 will result in an inverse with integer components. Students tend to be drawn to these projects, but sometimes I find it hard to push them further, such as requiring them to create their own encoding matrices, etc. Resources abound.
Modular arithmetic and row reduction - Uses matrix multiplication in modular arithmetic to encode and decode a message and row reduction in modular arithmetic to find the inverse. This is not that the decoded matrix is read using mod 26, but rather that the matrix operations are done with, say, mod 37. This project requires a little more tenacity on the part of the student, and this article by Keith Conrad discusses inverses of matrices under modular arithmetic.
Saturday, July 31, 2010
I'm a nerd. I freely admit it. Sometimes I see a result in mathematics that I think is fun, even if it is not immediately useful. I intended for this blog to be about applications of linear algebra, and by that I mean useful, if not a little esoteric, applications; applications that are juicy and interesting and have good visuals. Although the abstract applications of linear algebra to theoretical areas of mathematics are useful to someone, they do not have the hands-on feeling that I want. However, the nerd in me finds some abstract ideas sexy enough to include in this blog. The following is one of them. Although there is a connection between this idea and Magic Squares, which have constant row sums, I don't see an application right off. If you know of one, please let me know.
Theorem (already this post looks different than usual 'cause it has a theorem):
If an invertible matrix A had constant row sums of k, then the inverse of A has constant row sums of 1/k.
Proof (Oh, no. A proof. Just when this blog looked like it was just fun stuff):
Let A be an m-by-m invertible matrix with constant row sums of k. Let B be the inverse of A. Now, AB = I and the diagonal elements of I are all 1. Note that I has a constant row sum of 1. Hmmm. Let
(1)
be the elements of I. Then
(2)
is the sum of the elements of row i of I, and
(3)
is the sum of the elements of row n of A. Now, write the elements of a row of I = BA as the sum of products of elements of A and B:
(4)
Now, we're ready to put this all together.
Thus,
(5)
but the right-hand side of (5) is the sum of row i of B.
It seems like a trick, and in a way it is, but the trick is legit. What we have done is started with the sum of row i of I, written it in terms of the elements of A and B, and then seen that we could factor out the elements of B leaving row sums of A. To help you understand this, carefully write the elements of a general 3-by-3 BA in terms of the elements of B and A. Now, sum one of the rows of BA and rearrange so you can factor out the various elements of B. The sums of the elements of A left will each be a row sum. This wouldn't be a proof in general, but it should help you understand what is happening in that sum-switching step of the proof, and that it works because the elements of a matrix product are sums and then we sum a row of sums, and the terms within these two sums can be conveniently rearranged.
Questions:
We started with an invertible A with constant row sum. What if A had constant column sum of k instead of row sum? Will the inverse of A have constant column sum of 1/k as well? How would the proof go for that? This would be a great exercise in working with sums and indices.
This proof will not work if the constant row sum is k = 0. Can a matrix have an inverse if k = 0? If k is not zero, are we guaranteed that A is invertible? Can we determine if a magic square is invertible by the row sum alone?
Is there a relationship between the diagonal sums of a magic square and its inverse?
Is the inverse of a magic square a magic square? Is the inverse of a semi-magic square a semi-magic square?
Is the square of a magic square a magic square? Is the square of a semi-magic square a semi-magic square? Cubes? Fourth powers?
Friday, July 30, 2010
To the right we see a planar graph. It divides the plane into 5 regions which I have labeled A, B, C, D and E. We call region D a triangle because it has three sides. Regions B, C and D are quadrilaterals since they have 4 sides. Even though A is an infinite region in the plane, it has 3 sides and is called a triangle. The question we address here is whether we can draw planar graphs with all possible combinations of triangles, squares, pentagons, etc., or not. Also, can we determine which combinations are possible?
We will use the Euler Characteristic for the sphere to solve this problem. I know it looks like the figure above is drawn on the plane, but you can also think of it as drawn on a relatively flat part of a sphere. In this way, region A is not infinite, but we will still call the regions triangles, quadrilaterals, pentagons, etc., even though they now have a curve to them. The Euler Characteristic is V – E + R, where V = number of vertices, E = number of edges (not the region E above) and R = number of regions. The Euler Characteristic depends only on the surface on which the planar graph is drawn, and not the shape of the graph. For the sphere,
V – E + R = 2,
always. In the example at the above, V – E + R = 6 – 9 + 5 = 2.
Let's do a little counting. In the figure above, E = 9 is the number of edges. However, if we count the sides of the polygons we get 2 triangles x 3 sides each + 3 quadrilaterals x 4 sides each = 18 sides. This is twice as many as the edges, because polygon each side is counted twice for each edge, once for the polygon on one side of the edge and once for the polygon on the other side of the edge. For instance, in the edge count, E, the edge xy is counted for the triangle D and the quadrilateral C.
What about the vertices? In the figure above V = 6is the number of vertices. If we count the vertices of the polygons we get 2 triangles x 3 vertices each + 3 quadrilaterals x 4 vertices each = 18. In this case, we get three times as many polygon vertices as graph vertices because there are three polygons meeting at each vertex. For example, polygons A, B and C meet at vertex t.
These counting techniques and the Euler Characteristic will give us a system of equations for finding whether graphs with certain combinations of polygons are possible.
Example 1: Can we draw a planar graph with only triangles so that exactly three triangles meet at each vertex? If so, how many triangles will there be? We can answer this question with a system of linear equations. The first equation is the Euler Characteristic for the sphere:
V – E + R = 2.
Each region is 3 sided, but if we count 3R sides that will be double the edges since each side is counted twice:
3R = 2E.
Each region has 3 vertices, but if we count 3R vertices that will be triple the total vertices since three triangles meet at each vertex:
3R = 3V.
Solve this square system of 3 equations in 3 unknowns using your favorite method, and we find there is only one way to do this:
V = 4, E = 6 and R = 4.
The only solution is to have 4 triangles, 4 vertices and 6 edges as shown on the right, remembering that the outside region is a triangle. So, we could never draw a graph that had 5 triangles such that 3 triangles meet at each vertex. Give it a try to see why it can't be done.
Questions: Can we draw a planar graph of triangles where 4 triangles meet at each vertex? 5 triangles meet at each vertex? 6 triangles meet at each vertex? How would we change the system above to answer these questions. If the graph exists, try to draw it.
Example 2: What if there are two different types of polygons? Consider a graph of triangles and quadrilaterals, assuming that three polygons meet at each vertex. We will introduce two new variables: T and Q, the counts of the triangles and quadrilaterals, respectively. Now, the total number of regions is the sum of the two types of polygons,
T + Q = R.
Count the edges, 3 for each triangle and 4 for each quadrilateral, and as in Example 1, this counts each edge twice:
3T + 4Q = 2E.
Count the vertices, 3 for each triangle and 4 for each quadrilateral, and as in Example 1, this counts each vertex thrice, because 3 polygons meet at each vertex:
3T + 4Q = 3V.
Finally, we need the Euler Characteristic:
V – E + R = 2.
This time we'll use a matrix and row reduction to get the solution. The system is underdetermined, so we expect to get infinitely many solutions.
Sure enough, we have a free variable and we can write the general solution as
T = 12 – 2R, Q = –12 + 3R, V = –4 + 2R and E = –6 + 3R.
But in this application, the values of T, Q, V and E have physical meaning and must be positive. If V or E is zero, then the graph would be empty. We could assume that T or Q is zero, but we are interested in graphs with both triangles and squares. Now we can solve the inequalities below to see if there is a viable solution, and how many there are.
Okay, R is an integer and strictly between 4 < R < 6, so R = 5 is the only realistic solution to this underdetermined system. Now,
R = 5, T = 2, Q = 3, V = 6 and E = 9.
Draw this graph (don't forget that the outside region is one of the 5 regions and must be either a quadrilateral or a triangle). The graph is at the bottom of this blog, but don't peak before you give it a try.
Questions:
1. Can you draw a planar graph with pentagons and hexagons such that three polygons meet at each vertex? If so, how many of each polygon are there? Can you draw them?
2. Can you draw a planar graph with triangles and quadrilaterals such that four polygons meet at each vertex? I have written the equations and solved the system for this case, and this may have infinitely many solutions, but I haven't had the time to draw more than two of the solutions and would like to see an algorithm for drawing all of them.
3. Other surfaces, such as a torus (donut) have different Euler Characteristics. How does one draw a graph on a torus? What are the solutions to the questions above if the graphs live on a torus? Wolfram MathWorld has a list of the Euler Characteristics for surfaces, but WikiPedia has nice images of those surfaces if you scroll to the bottom of the article.
To limit this blog to a few pages, a lot is left unsaid. But again, these posts aren't meant to give an in-depth discussion of the topic, but just an introduction. Go exploring for more about this topic. | 677.169 | 1 |
Elimination theory and resultants are important manipulative tools that contribute theoretical insights and practical algorithms to computer graphics and geometric modeling. In this tutorial, resultant properties are listed to enhance overall understanding of resultants. For bivariate resultants, two explicit expressions are presented: the Sylvester matrix and the Bezout determinants. | 677.169 | 1 |
Subject Material: We will cover chapters 1 - 6 and parts of 7 of
the text. For a more detailed list of chapter sections, see the
homework page.
Lecture: Attending the lecture is a fundamental part of
the course; you are responsible for material presented in the lecture
whether or not it is discussed in the
textbook. You should expect questions on the exams that will test your
understanding of concepts discussed in the lecture.
Reading: Reading the sections of the textbook
corresponding to the assigned homework exercises is considered part of the
homework assignment; you are responsible for material in the assigned reading
whether or not it is discussed in the
lecture. You should expect questions on the exams that will test your
understanding of concepts addressed in the reading and assigned homework
exercises.
Homework: See the
homework page for guidelines, assignments
and due dates. Your homework grade will be based on your best eight (8) of nine
(9) graded homework assignments. You should make every effort to complete each
assigned homework problem. You may seek help during office hours and/or section
with any exercises you have difficulty solving. Students are also encouraged to
discuss homework with fellow classmates. However, each student is expected to
write up his/her own solutions independently.
Late Homework: Late homework is not accepted.
Electronic Computing
Devices: Graphing calculators and computer programs (or
online computing websites such as Wolfram|Alpha) can be very helpful when
working through your homework. However, a calculator/computer should be used as
an aid in the learning concepts, not just as a means of computation. You should
use these devices when working on math problems at home, but always keep in mind
that you will not be allowed access to any electronic computing devices during
exams. Of course, this also means that you will not be asked to solve problems
on exams that require the aid of an electronic computing device.
The use of electronic devices will not be
permitted during exams.
You must take the midterm
on the scheduled dates and at the scheduled time.
You cannot take any exam early or late.
You must bring photo ID with you to all the
exams.
You will not be allowed to use a calculator
during the midterm exams.
If you violate the instructions of a midterm
or communicate in any way with any other student during a
midterm, you will receive a zero on that exam, and the zero will
not be dropped when calculating your cumulative course average.
The Final Exam is
cumulative. You cannot take the final exam early or late. It
is your responsibility to ensure that you do not have a schedule conflict
involving the final exam. You should not enroll in this class if you cannot sit
for the final exam at its scheduled time.
You must bring photo ID with you to all exams.
You will not be allowed to use a calculator on
the final exam.
If you violate the instructions of the final
exam or communicate in any way with any other student during the
final exam, you will receive a zero on the final exam.
Regrades: The midterm exam and graded homework assignments
will be returned in the discussion sections. If you find a grading or point
totaling error on an exam or a homework assignment,
you must return it immediately to your
TA. Regrade requests will not be considered once the exam or assignment
leaves the room. If you do not retrieve your graded exam or assignment during
discussion section, you must arrange to pick it up from your TA within one week
after it was returned in order for any regrade request to be considered.
Grade Recording Errors: Keep all of your returned homework and exams and
check TED to make sure that the grades on your papers are the same as the grades
recorded on TED. If there is any mistake in the recording of your scores, you
must bring us the original assignment/lab/exam in order for us to make a change.
Grades: Your cumulative average will be the best of
the following two weighted averages:
midterm1 (25%) + midterm2 (25%) + final (35%) +
homework (15%)
best midterm (25%) + final (60%) + homework (15%)
After your weighted average is calculated, letter
grades above scale to be more lenient (depending on the overall class
performance), but we guarantee that we will not adjust the scale to make it
harder to get a better grade. Please note:
You must pass the final examination in order
to pass the course.
Since there are no makeup exams, if you miss a midterm exam, then your course grade will be
computed with the final exam counting 60% of your weighted
average.
Piazza: Piazza is an online discussion forum that allows
you to ask questions using mathematical symbols and expressions. Piazza was
designed to enable you to get help quickly and efficiently from classmates, TAs,
and instructors. Rather than emailing questions to the teaching staff, you are
encouraged to post your questions on Piazza. Find our class page at:
piazza.com/ucsd/Fall2015/math120A/home. If you have any problems
or feedback for the developers, email: team@piazza.com.
Suggestions: Below are some suggestions that I hope will help
you to succeed in this course:
The best study strategy for this class is to
go over all material covered in lectures and all HW problems and
then if there's time, do additional problems from the textbook.
For the final you should also study all the problems done in the
midterm.
Spend sufficient time on the
course. According to the policy of UCSD's Academic Senate, "The
value of a course in units...shall be reckoned at the rate of
one unit for three hours' work per week per quarter on the part
of the student." Since Math 120A is worth 4 credits, you should be willing to spend about 24
hours per week on the course.
Keep up with the homework and do not miss a
midterm. Missing a midterm puts a lot more pressure on your
final exam performance .
Get started on the homework assignments
early. This will enable you to make the most of your discussion
section time by coming prepared with specific questions.
Academic Dishonesty: Academic dishonesty is considered a serious
offense at UCSD. Students caught cheating will face an administrative sanction
which may include suspension or expulsion from the university. It is in your
best interest to maintain your academic integrity. (Click
here for more information.) | 677.169 | 1 |
Prerequisites: Math ACT 27 or higher or a grade of at least C- in Math 1250 or dept consent; Credit will not be granted if credit has been received for: 1296, 1596
Grading: A-F only
Liberal Education Category:
This course satisfies the UMD Liberal Education requirement for Category Two: Math, Logic, and Critical Thinking. Calculus is a universal mathematical tool that is used in many diverse areas including business, economics, biology, geology, chemistry, physics, and engineering. Whenever measured quantities change with respect to time, or other variables, calculus is probably involved. This course develops the fundamentals of calculus suitable for applications in the life and earth sciences. By the end of the term, the successful student should understand the importance that calculus plays in modeling real-world phenomena by constructing and analyzing numerous models selected from ecology, wildlife/fisheries management, epidemiology, physiology, groundwater diffusion, and seismic phenomena.
Course Description:
The course will cover standard topics in differential calculus, integral calculus, and introductory differential equations. Topics include limits, continuity, derivatives and applications of derivatives; integration, the fundamental theorem of calculus, integration techniques; differential equations. Applications to biology are used throughout the course. The material is mostly covered in Chapters 1-5.3 of the Adler text. Some supplemental material, not included in the text, may occasionally be presented in lecture.
Comparison to Calculus I, Math 1296: Roughly 80% of the material in the two courses is the same. Math 1290 skips a few 1296 topics and covers others in less depth. Math 1290 includes an introduction to differential equations. In addition, the applications in Math 1290 focus on biology and ecology, while in 1296 applications are chosen from a variety of areas of science and engineering. (Students who take Math 1290 will be able to register for Calculus II (Math 1297) if they wish to continue on in Mathematics.) In summary: Math 1290 covers the parts of Calculus most necessary to allow the inclusion of an introduction to differential equations.
Text: Modeling the Dynamics of Life 2nd Edition by Frederick Adler published by Brooks/Cole, 2005 | 677.169 | 1 |
Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
Click on each unit below to access course work for the unit.
Measurement, Volume and Surface Area
TIMING
13 periods
OVERVIEW
Students will become familiar with metric and imperial conversions. Students will review determining perimeter, surface are and volume of a variety of shapes as well as reviewing the Pythagorean theorem. Students will then apply these skills as they solve real world problems through both collaboration and individual work.
OVERALL
EXPECTATION
Measurement and Trigonometry 3: solve problems involving the surface areas and volumes of three-dimensional figures, and use the imperial and metric systems of measurement
SUMMATIVE
Assignment Series: Solving Real Life Problems Involving Perimeter, Area and Volume(included in 70% term mark)
Students will remember skills and concepts about linear relations from grade 9 including: slope, rate of change, y-intercepts, creating table of values, recording equations of lines. They will apply their knowledge in a variety of real world contexts and then master their pure math skills relating to lines.
Student will explore this new concept of parabolas and quadratic relations. They will make connections between graphs and equations of parabolas and develop new algebraic skills in writing the equation of quadratic relations in different forms. Students will further their exploration and understanding of parabolas and quadratic relations in grade 11 math.
Students will use their skills of ratio and proportion and apply them to solve similar triangles. Student will complete an investigation using a variety of techniques to determine unknown heights by comparison with similar triangles. Students will communicate their findings in a written report.
OVERALL
EXPECTATION
Measurement and Trigonometry 1: use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity
SUMMATIVE
Report: Using Similar Triangles to Determine Unknown Heights(included in 70% term mark)
Trigonometry
TIMING
12 periods
OVERVIEW
Students will discover the relationship between the sides of right angled triangles and become familiar with solving right angled triangles using the primary trigonometric ratios. Students will explore real world problems and research various careers that use trigonometric skills. Students will further their exploration and understanding of trigonometry in grade 11 math.
OVERALL
EXPECTATION
Measurement and Trigonometry 2: solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem | 677.169 | 1 |
Are you looking for a math literacy book that is different from those available now? Are you looking for any algebraic literacy textbook?
These books get written by people who want to teach the courses. We understand the goals of the courses, the type of content that should be present, and how to present this material so that students can succeed.
Perhaps you are somebody who might be interested in writing either a Math Literacy or an Algebraic Literacy text … either by yourself or as part of a writing team. If so, you can certainly approach any publishing company to start the process.
In particular, Pat McKeague of XYZ Textbooks is willing to work with potential authors of textbooks for our new courses. He is excited about developing more textbook choices for us, while providing materials to students at a lower cost. When XYZ publishes a textbook, they do some of the wrap-around work (such as videos).
I appreciate Pat's support of our work and his willingness to work with authors. If you are interested in learning more, contact Pat at pat@mckeague.com
The textbooks should be a close approximation to the course goals & outcomes:
Clearly, the intent is that any textbook focus on understanding and reasoning. The level of "context" and "small group work" can vary (though always being a part of the package); some of this could be left up to the instructor using the materials.
Share this:
Within our mathematics community, much of our recent efforts have been directed at presenting students mathematics related to problems (contexts) that are likely to be important to them. Some curricular work is limited to the mathematics for which such a context can be presented. Although relevant context is helpful, we lose something important when the context becomes more important than 'mathematics'.
In some ways, this makes sense … Mathematics has always had roots deep in practicality.
However, I see two failures resulting from these approaches:
Mathematics is not always practical when ideas are developed or discovered.
General education seeks to go beyond the parochial.
In the American culture of 2016, we seem to validate the notion that "I only have to care about things that impact me directly." When we honestly tell students that this mathematics is important even though we are not showing a context for it, we should be able to expect students to honor the statement. In many ways, learning mathematics without context is a good training program for employment … I suspect that the majority of workers work in a job with little innate value to them, in which they need to honor a supervisor's statement that doing a job a certain way is important.
The role of general education has been both integral to higher education and marginalized in higher education. The values of 'different perspectives' and 'modes of thought' represent the building of capacity in a society to think about difficult problems without resorting to slogans and over-simplifications. When general education works, it is a beautiful thing. This type of rising to a higher level of problem solving can not occur when the classroom is limited to the shared current concerns of those present.
If we truly believe that students are well-served by allowing them to focus on their own interests and concerns, sure … let's limit their mathematics to contexts that they can understand at the time.
I think that limitation is a dis-service to students (and is not respecting mathematics as a set of disciplines). Sure, we can have lots of fun when students are enthusiastic about our work in class. Do they have any better notion of what 'mathematics' is? Did the experience result in anything more than a few concepts that are applied in concrete ways?
Our courses should always contain significant elements of what I call "beautiful and useless mathematics". "Beautiful" refers to the aspects of mathematics which appeal to mathematicians … which can vary from person to person, and from one domain to another. "Useless" refers to the ideas being developed in an abstract way without knowing if there will ever be any practical use.
One example of such 'beautiful and useless mathematics' would be functions which have a rate of change equal to the function. The number e is not immediately reasonable when we deal with concrete multiplicative change. We can contrive some contexts where the base e can be used, though most of these are more accessible to students using a percent growth rate (or decay). The use of e for the function, and for the rate of change in the function, is a thing of beauty.
In some ways, this post boils down to this statement:
Don't sanitize any mathematics course to the point where all artistic merit is destroyed.
Although this post relates to a recent post on 'where STEM students come from', I think the idea is valuable for every student who walks in to a math class. We are not mathematicians because it is practical (though it is); we are mathematicians because there was something that attracted us. Our students deserve to see at least a small corner of the wonderful canvas called 'mathematics'. | 677.169 | 1 |
Homework for Tuesday
Linear equations homework
and begin working on the Daily Journal
and CD assignment.
July 19 - Tues
Classwork Linear equations: Egypt Rhind Papyrus, Babylon, China:
Egyptian Multiplication/Division The origin of algebraic manipulation via
Linear Equations in One Unknown - Islamic Method worksheet from the
CD.
Chinese method of double false position of solving two linear equations
with two unknowns via Chinese Problems worksheet from the CD.
System of Three Linear Equations - The Chinese Solution
from Algebra Activities from Many Cultures Multicultural Origin of the Quadratic Formula
worksheet from
Algebra Activities from Many Cultures Brief history of the Pythagorean Theorem via web search:
site
and
site MacTutor History
of Mathematics Archive Scarecrow's Theorem
Slide 1Slide 2Copyright Slide Cubic equations: Europe - leading to sqrt(-1)
Solutions to the quintic
Maple demo and fictional account of Galois
Begin Wednesday's classwork.
Homework for Wednesday
Daily Journal reflections for the first two days should be completed.
Your CD project topic must be approved by Dr. Sarah by
Wednesday.
July 20 - Wed
Classwork
What is geometry? Groups discuss and come back together.
Brief history of early geometry knowledge including the
Babylonians, Egyptians, Greeks (Thales, Pythagoras, Plato, Euclid),
Chinese, and Africans.
Mention the Timeline for Geometry: Multicultural Highpoints
from Geometry Activities from Many Cultures Skim LAV 6 section of the CD
on Origins of Length, Area, and Volume Measurement
Mention CD LAV 80 on Thales' Shadow Measurement Activity
Mention CD
Trig 80 and LAV 88 on Eratosthenes' Size of the Earth and
Trig 112 on Al-Biruni's Calculation on the Size of the Earth
History of Euclid's elements including
Euclid's 5th postulate (prop 29 and later).
Sum of the Angles in a Triangle:
Write out
Proposition 32 in modern language.
Discuss Euclid's historical proof,
paper folding,
and then an
Intro to Geometric Constructions
via a
construction in Sketchpad.
Dr. Sarah models the construction.
Discuss the history
(attributed to Pythagoras),
and compare and contrast the activities,
discussing the benefits and difficulties with using them in various classes.
Then the group goes to the computer lab to try the Sketchpad construction.
Pythagorean Theorem:
Discuss the Yale tablet and Babylonian Pythagorean knowledge.
Discuss the history of
Euclid Book 1 Proposition 47 and of it's importance in measurement.
Discuss Sketchpad demonstrations of the Pythagorean Theorem. From
Sketchpad, under File/open, go to
Desktop/205Math(yourcomputersnumber)/Applications(MacOS9)/Sketchpad/ Samples/Sketches/Geometry/Pythagoras.gsp
Go through Behold Pythagoras! and Puzzled Pythagoras.
Return to the classroom.
Highlight the other proof from the video
(Euclid's Pythagoras), and then Dr. Sarah's models the construction in
Sketchpad.
Mention CD activities
LAV 124 Pythagorean Theorem,
LAV 130 Practical Pythagorean Theorem,
Geom 103 Pythagorean Project
Hand out A Chinese Proof of the Theorem from
Geometry Activities from Many Cultures
in order to complement the Sketchpad Behold Pythagoras! activity
Compare and contrast the methods used in the classroom.
Applications of series:
history of the approximation of Pi
and the number of known digits of Pi.
Apu and pi in Maple, and search for the world record on
the number of digits. Discuss Bailey and Borwein's series. | 677.169 | 1 |
Free Student aptitude test, this topic is mainly used to preparation for campus recruitment program. It involves several topics. But here we can
revise the few topics. The topics are arithmetic and geometric progression and analytical geometry. Analytical geometry involves the main topics such as find the equation of the line and slope of
the line. These are revised problems with solutions.
Free student aptitude test with solutions
Free student aptitude test Example 1:
Prove that the sum of n arithmetic means between two numbers is n times the single
A.M between them.
Solution:
Let A1, A2… An be the n A.M's between a and b.
From the example
A1+ A2 + A3 + … + An = (n(a+b))/2
= n × (A.M between a and b)
= n (single A.M between a and b)
Free student aptitude test Example 2:
If A(-2,5) and (4,-5). Then solve the equation of the locus of a point. Given
Statement PA2 – PB2 = 20.
Solution:
A(− 2, 5) and B(4, − 5) are the two given points. Let P(x1, y) is any point on the locus.
Given that PA2 – PB2 = 20. | 677.169 | 1 |
Big Ideas: Many problems can be solved arithmetically or algebraically. An algebraic solution is simply a generalized arithmetic solution. This lesson builds on students' knowledge of solving real-world word problems with arithmetic, and writing and sol... | 677.169 | 1 |
Math 99 Problem Set Policy:
Your main efforts in this class will center around 6 problem sets. The
rules for these are covered in the Mathematics Department Statement on
Problem Sets and Academic Honesty, which will be handed out to you when the
first set is assigned. The student form should be signed and returned
with your solutions to the first problem set. The problem sets are to be
worked independently; that means you should not discuss them outside of
class except with the course instructor.
Problem sets will be assigned each Friday, and will be collected on the
following Friday. You should begin work on them as soon as possible, as
many problems will require that you think about them more than once.
Starting on a problem set does not mean just beginning the first problem
or two and leaving the rest for later; you should get in the habit of
reading through all the problems, and working on them a little bit
each day. The harder problems frequently come at the end, so it is not in
your best intrest to leave these until the night before they are due. One
of the reasons for giving you a week to work on the problems is so you can
ask questions about them during class; be sure to take advantage of this by
starting on all the problems early.
Please staple your homework pages together, and trim the left
edge if they are torn out of a spiral notebook. Unstapled or ragged
homework will not be graded, nor will homework that is turned in late.
Please print on only one side of the paper, clearly and legibly. You
should not turn in your initial draft of your answers, but should copy out
a final draft once you have the methods worked out. I will not grade
material that I consider to be a first draft.
Since this course deals extensively with the style and technique of writing
mathematics, you should be especially careful to write clearly and
carefully on the work you turn in. That means you should write in
complete, english sentences when you write up your homework.
Mathematical notation is shorthand for english words, so you can
incorporate these into your sentences, but they should read as a sentence,
and you must explain what you are doing as you do it. I will give
you examples of this as we go.
Copies of the best solutions will be made available in a notebook
outside my office. | 677.169 | 1 |
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Category: Algebra
After a week of attending Mathnasium she could completely understand the problems and get the answers with confidence. If you don't see the activity on the screen, you may need a program called Adobe Acrobat Reader. Every unit closes with a section of Assessment Readiness that features selected response, constructed response, and extended response items. This method is one way to derive the quadratic formula. Excellent resource for taking the AP Calculus exam. [more] spiritedmama 09/07 Barron's AP Statistics, Grs. 9-12 -- $5 ppd.
ADDMULT2: This program is less complicated to use. Before that I had okay teachers, nothing marvelous, but they couldn't devote all their time to math, so it wasn't as good as it was now. If you miss both midterm exams or if you miss the Final Exam, you will fail the course. Text book and Student solutions manual ISBN 9780321785237. Practice can continue as long as desired in a non-threatening format which helps build self-esteem and confidence.
Thank you, Mathnasium for keeping "math fresh" for my 6th grade son! Working together, we should implement the 21st century version of the Second Amendment: Everyone shall have the right to bear "mathematical arms"—to possess mathematical knowledge and tools needed to protect us from arbitrary decisions by the powerful few in the increasingly math-driven world. Consider joining us for our Preconference, a topic-intensive interactive all-day workshop on Thursday, February 23, 2017 from 9 am - 3 pm.
By the time you finish reading them, however, you'll be a whiz at tons of pre-algebra topics, including integers, negative numbers, absolute value, inequalities, the distributive property, working with variables, word problems, exponents, functions, graphing, and tons of ways to solve for x. MathGames by TeachMe - Practice Makes Permanent! If there are irrational or imaginary/complex roots in addition to rational ones, the program gives you the coefficients of the remaining polynomial.
In this essy i will state my opinion on whether I am good or bad at math, my good and bad experiances with math, and what I wish to accomplish this year in math. Click Here To learn more about the instructor in the math videos. We will be holding a two-day event at the College of New Jersey on July 20th and July 21 from 9 AM to 12 Noon each day. He taught really well and I had a blast learning math with him. This year i want to strengthen my math skills so i feel more confident, Also be more organized with my work instead of just writing on the page to get the answer.
Math is usually harder because i rush through my work a lot and I do not bother to double or triple check my work. Unfortunately many textbooks go straight to the rules, procedures and formulas, forgetting that these are real life problems being solved. This program factors polynomials of one variable. Also contains continuous compound interest in the form of Pe^(rt). Providence, RI: American Mathematical Society. Where does carbon come and go in the Earth system?
The explanations often highlight ideas on best problem solving approaches, which is something you don't usually see in regular algebra textbooks. Also, I was asked after class if I could type up the assignments and make them available on the webpage. Jacobs' algebra is also on the easy side, as far as CONTENT goes. I enjoy learning new things in math class and always took notes on it to prepare myself for a quiz or test. In all areas, similar functionality is provided for real and complex matrices, in both single and double precision.
Can you tell I love the Frayer Model for vocabulary? I hated it, because most of time I didn't understand the problem, or I just didn't try to do it. This assignment is designed for completion by the students at home. Any other use (including any commercial use) requires an explicit permission from the author. Subtracting and simplifying linear expressions with multipliers. eMathInstruction was founded on the simple premise that 21st century technology can and will change the entire landscape of mathematics education.
Bt if i don't live on rent, they will give this allowance? Mathematics is commonly called Math in the US and Maths in the UK. They say grades are often much lower than middle school since the topics we learn in high school is harder and more confusing. CCSS Algebra 1 Subscription Samples: All sample materials below are to be used only by individual teachers in their face-to-face classrooms. Factoring From rags to riches - climb the $1,000,000 ladder.
The first exam on Friday will cover all the material from chapters 1 and 2. Any student with a College Algebra Exemption has satisfied the University's College Algebra General Education requirement. Download MathPapa - Algebra Calculator apk 1.0.5 and all version history for Android. There are a few different ways to use this book: You can skip directly to the chapters that will help you with tonight's homework assignment or tomorrow's test. | 677.169 | 1 |
Mathematics (4024)
Mathematics (4024)
This course places emphasis on broad study across a wide range of subject areas which includes the development of their mathematical knowledge of numbers, patterns and relationships. Also, it enables students to solve problems and interpret results.
· Numbers
· Limits of accuracy
· Straight line graphs
· Angle
· Set language and notation
· Ratio, proportion and rate
· Algebraic representation and formulae
· Locus
· Function notation
· Percentages
· Algebraic manipulation
· Mensuration
· Squares, square roots, cubes and cube roots
· Use of an electronic calculator
· Solutions of equations & inequalities
· Trigonometry
· Directed numbers
· Measures
· Indices
· Statistics
· Vulgar and decimal fractions and percentages
· Personal & household finance
· Graphical representation of inequalities
· Probability
· Ordering
· Money
· Geometrical terms and relationships
· Matrices
· Standard form
· Time
· Bearings
· Transformations
· The four operations
· Graphs in practical situations
· Geometrical constructions
· Vectors in two dimensions
· Estimation
· Graphs of functions
· Symmetry
·
Prior Learning
It is recommended that students should have previously studied an appropriate lower secondary Mathematics program. | 677.169 | 1 |
Question:Answers:Lets face it. An answer key is nice but it does not teach you the math. Nor does a solutions manual when you are old enough that they come with your math books. Learn the material. Math builds and builds and builds and. | 677.169 | 1 |
Basic Mathematics
by Ginny Crisonino & Steven Slavin
Basic Mathematics by Ginny Crisonino and Steve Slavin has a simple pedagogy: You learn math by doing math. The authors start by showing students how to solve a problem, and then ask students to solve similar problems. Students get positive reinforcement by working their way through the text, checking their work, and correcting their own mistakes, all while mastering the material.
Each chapter has a simple presentation style with an identical format. A section begins with a few illustrative examples, followed by a problem set for the student to work on with full solutions, and finally a workbook style set of exercises which can be done in class or as homework. The step-by-step solutions to the odd numbered problems are given in the back of the book. This format allows students to work their way through the book and learn the basic foundation of mathematics. | 677.169 | 1 |
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's. This all-in-one-package includes more than 1,100 1,105 fully solved problems Concise explanations of all calculus concepts Expert tips on using the graphing calculator This all-in-one-package includes 73838 fully solved problems The latest course scope and sequences, with complete coverage of limits, continuity, and derivatives Succinct explanation of all precalculus concepts88 fully solved problems Succinct review of physics topics such as motion, energy, fluids, waves, heat, and magnetic fields Support for all the major textbooks for physics for engineering and science courses Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores!
See how to solve precalculus problems with this enhanced ebook that features 30 videos of professors working through solutions! Confusing textbooks? Missed lectures? Not enough time? Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. And there are plenty of problems for you to practice on, with more than 700 precalculus problems with fully worked solutions so you can check your work, or get help when you need it. Plus, this new enhanced edition features video solutions of professors showing exactly how to solve problems. If you want top grades and a thorough understanding of precalculus, this powerful study tool is the best tutor you can have!
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately for you, there's Schaum's. More than 40 million students have trusted Schaum's Outlines: Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's. This all-in-one-package includes more than 600 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 618 fully solved problems to reinforce knowledge Concise explanations of all trigonometry concepts Updates that reflect the latest course scope and sequences, with coverage of periodic functions and curve graphing. Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores! Schaum's Outlines--Problem Solved.
Schaum's has Satisfied Students for 50 Years. Now Schaum's Biggest Sellers are in New Editions! For half a century, more than 40 million students have trusted Schaum's to help them study faster, learn better, and get top grades. Now Schaum's celebrates its 50th birthday with a brand-new look, a new format with hundreds of practice problems, and completely updated information to conform to the latest developments in every field of study. Schaum's Outlines-Problem Solved More than 500,000 sold! This thorough review of standard college courses in trigonometry has been updated to reflect the latest course scope and sequences. The new edition includes expanded explanations of the aspects of each periodic function, and updated information for the curve graphing section.
Improve your understanding of calculus and your grades will also improve with this effective study aid as your guide. Comprehensive and packed with 272 solved and supplementary problems, it can be used with any undergraduate calculus textbook.
Confusing Textbooks Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of | 677.169 | 1 |
A non-programmable calculator will be allowed for use in Section III only. Please note that your calculator will not be permitted if it has any of the following
o A button called PRGM or prog (or similar);
o A graphics function;
o Keys or buttons with alphabetic letters associated with them (allowing letters and words to be entered).
My calculator is a Casio fx-100au. Now it obviously does not have the first two things that are not permitted but the last point talks about letters. It has the letters A, B, C, D, E, F, X, Y, M. I'm not even sure what they are used for. Are they going to care about this? I have a couple of other scientific calculators at home but they all have some sort of letterings on them. | 677.169 | 1 |
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Some resources used in my classes….
Below I have given details of some resources I am currently using with my classes or have recommend to my students so they can explore examples further themselves.
Year 12 (age 16-17) I want to talk about quadratic inequalities this week so I thought I'd use the Desmos graphing calculator to draw some pictures! Click on the image for the Desmos page and select 'projector ' mode for display on the interactive whiteboard. Last week a student in this class asked where she could find some additional resources on polynomial division. Note: I use the Desmos calculator so much I have decided it deserves a page of its own here (under Resources).
Year 13 (age 17-18) Some students in a Further Maths class asked for some Polar coordinates resources to support their studies – so a post for them on my blog for students – these resources would also work well on the interactive whiteboard for use in class. As regular readers know I am a great fan of WolframAlpha and use it with all my classes (WolframAlpha now have a paid for service but it is still completely free to use to check answers for an unlimited number of queries, the free use limits step by step solutions to 3 a day). One of this class showed me that he has the WolframAlpha app on his phone.
Year 11 (age 15-16) My Year 11 group are studying the AQA Level 2 Certificate in Further Mathematics (a course I am very much enjoying teaching) as well as completing their GCSE course this year. We have been studying calculus and I have found the Desmos graphing calculator very useful to illustrate problems we have been solving. This class have mock examinations coming up and I wanted to recommend some additional resources for them (we have various texts and the AQA support is excellent but the more the better and there is currently no textbook for the course); one site with some very useful resources for some parts of the course such as introductory Calculus is David Smith's 'The Maths Teacher.'
Year 8 (age 12-13) I have a Year 8 class this year, none of whom I have taught before, we have been looking at surface area and volume. Math Open Ref has a rather nice animation which helps when looking at the surface area of a cylinder. (More on John Page's Math Open Ref). I will also use this site when we look at constructions soon. Most had not seen WolframAlpha before so were quite impressed at how easy it is to check working! There are slideshows available for students showing the syntax for a selection of examples on my blog for students. | 677.169 | 1 |
Mathematics 1. Basic Concepts.
Description
MATHEMATICS - BASIC CONCEPTS is a course for studying Matemáticas in English. The language and content have been carefully adapted to reflect the abilities of Secondary students. These books present the key contents of each level in a simple and practical format, acompanied by abundant execises and worked answers.show more | 677.169 | 1 |
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Summary
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's DEVELOPMENTAL MATHEMATICS FOR COLLEGE STUDENTS, Third Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, algebra is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the "language of algebra," the text's fully integrated learning process is designed to expand students' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. | 677.169 | 1 |
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Algebra I (Core)
Grade Level: High School
Subject: Math
Type: Core
In this course, students explore the tools of algebra. Students learn to identify the structure and properties of the real number system; complete operations with integers and other rational numbers; work with square roots and irrational numbers; graph linear equations; solve linear equations and inequalities in one variable; solve systems of linear equations; use ratios, proportions, and percentages to solve problems; use algebraic applications in geometry, including the Pythagorean theorem and formulas for measuring area and volume; complete an introduction to polynomials; and understand logic and reasoning. Prerequisites: Pre-Algebra (Core) or equivalent | 677.169 | 1 |
Bbc bitesize maths coursework
When you master our gcse maths quizzes, you'll feel like a huge pile of angles and sequences have been lifted from your shoulders. Start writing remarkable essays with guidance from our expert teacher team learn more international baccalaureate in the vast majority ofschools gcse maths. For example, in english, a pupil may have to complete 4 pieces of coursework, each over a thousand words long gcse maths intermediate papers. Interactive, animated revision help from the bbc, with support from the tv series. About bbc bitesize helping 5-16 year olds with their coursework the additional maths content – up to 20% of the marks in the chemistry exam paper will.
Maths learning resources for adults, children, parents and teachers organised by level and topic. This activity requires students to read descriptions of different sampling techniques and decide which situations match with most suitability. Physics physics is the study of energy, forces, mechanics, waves, and the structure of atoms and the physical universe in this section of gcse bitesize you'll find. Find and save ideas about edexcel gcse maths on pinterest bbc bitesize - gcse maths (2015 onwards) ict as coursework help.
Maths ks4 how is the course assessed %coursework / examination etc in gcse mathematics, 100% of the course is assessed through examination at the. It is time to start thinking as an adult and get the best gcse coursework writing help on the market our team of experts is always ready to help you. Exam board content from bbc bitesize for students in england, northern ireland or wales choose the exam specification that matches the one you study. Math's coursework: pythagoras triples maths number patterns investigation however, because 4 are the difference of the difference, the formula must. A correlation coefficient is a number between -1 and +1, which measures the degree to which two variables are linearly related maths estimation coursework. | 677.169 | 1 |
IMPORTANT
ANNOUNCEMENTS:1) Sections will not meet on
Monday, March 28. 2) Please read chapters I-II (Pollard &
Diamond text) before the first lecture. These are just a review of
part of the theory of quadratic fields and their rings of integers,
covered in Math 104B (Winter 2005), and will be assumed as known. A
brief review of this material will be given in lecture, on Monday 28.
COURSE DESCRIPTION
WHAT IS MATH 104C? This is the third
undergraduate course in number theory. In this course we examine topics
from algebraic number theory and focus on the following major themes:
1) Fields of algebraic numbers (a.k.a. Number Fields); 2) Rings of
algebraic integers; 3) Arithmetic in rings of algebraic integers
(integral bases, ideal theory, ideal-class groups etc.); 4)
Applications to Diophantine equations.
TEXT H. Pollard and H.G.
Diamond, The theory of algebraic
numbers (Third, revised edition); Dover Publ. 1998. You
are expected to read the text BEFORE each lecture 1+2: 20% each; Final Exam: 40%;
Homework: 20%. The grading will be done on a curve, the median
corresponding to a B-/C+. | 677.169 | 1 |
Category: Precal c 5 A-N Algebraic Reasoning
(5) Algebraic reasoning. The student uses process standards in mathematics to evaluate expressions, describe patterns, formulate models, and solve equations and inequalities using properties, procedures, or algorithms. The student is expected to:
(A) evaluate finite sums and geometric series, when possible, written in sigma notation;
(B)represent arithmetic sequences and geometric sequences using recursive formulas;
(C) calculate the nth term and the nth partial sum of an arithmetic series in mathematical and real-world problems;
(D) represent arithmetic series and geometric series using sigma notation;
(E) calculate the nth term of a geometric series, the nth partial sum of a geometric series, and sum of an infinite geometric series when it exists;
(F)apply the Binomial Theorem for the expansion of (a + b)^n in powers of a and b for a positive integer n, where a and b are any numbers;
(G)use the properties of logarithms to evaluate or transform logarithmic expressions;
(H) generate and solve logarithmic equations in mathematical and real-world problems;
(I)generate and solve exponential equations in mathematical and real-world problems;
(J) solve polynomial equations with real coefficients by applying a variety of techniques in mathematical and real-world problems;
(K) solve polynomial inequalities with real coefficients by applying a variety of techniques and write the solution set of the polynomial inequality in interval notation in mathematical and real-world problems;
(L) solve rational inequalities with real coefficients by applying a variety of techniques and write the solution set of the rational inequality in interval notation in mathematical and real-world problems;
(M) use trigonometric identities such as reciprocal, quotient, Pythagorean, cofunctions, even/odd, and sum and difference identities for cosine and sine to simplify trigonometric expressions; and
(N) generate and solve trigonometric equations in mathematical and real-world problems. | 677.169 | 1 |
This supplemental resource is the first of its kind that is solely dedicated to Mathematical Modeling!
The Common Core Modeling Books are designed to facilitate teaching and learning of essential common core state standards concepts and skills through mathematical models. This strategy increases the student's ability to understand and solve problems by choosing and using models that represent, justify, and/or explain problems and solutions, strengthening conceptual understanding.
For students to become mathematically proficient, they must be able to experiment with representing problems in multiple ways including numbers, mathematical language (words), drawing pictures, using objects, making a chart, list, graph, or creating equations. Students need opportunities to gain conceptual understanding of these representations (models) to choose and use them appropriately.
ACALETICS®Common Core Modeling Books for Grades 3-5 cover a number of Common Core State Standards at each grade level. These books are ideal for supplementing class work, Math ClubSM, and HomePrepSM. | 677.169 | 1 |
Reading Level 3 - 4Interest Level 6 - 12
AGS Mathematics series offers text that is high-interest, low-readability, which makes it easy to engage students who struggle with reading, language, or a learning disability. Includes full-color photos, illustrations, and examples throughout.
The students who would benefit from these textbooks are those who:
divide their time between regular classrooms and sheltered environments.
read below grade level.
need dedicated support to make lessons understandable.
may move directly to work or transition programs.
Algebra Provide students with all of the concepts and skills they need to succeed in a first-year algebra course. Correlated to NCTM Standards, the content provides students of all abilities with essential preparation in problem solving, calculator usage, and application lessons that demonstrate how algebra is integrated with related content areas such as geometry, probability, and statistics.
Teacher's Resource Library CD-ROM - offers hundreds of activities, the Student Workbook, a Self-Study Guide for students who want to work at their own pace, two forms of chapter tests, plus midterm and final tests.
System Requirements for CD-ROMs:
Acrobat Reader 4 or 5 - requires 8MB RAM or 64MB RAM, respectively, to install from resource cd. This step may not be necessary if Acrobat Reader 4 is currently installed on your computer. | 677.169 | 1 |
(i) help formulate, solve and analyze two-scale or
multiple-scale problems; and
(ii) permeate various scientific
disciplines.
Applications and examples are intended to span
fluid dynamics, condensed matter physics, materials science,
electrical and mechanical engineering, biology, astrophysics,
applied probability, and number theory. There are two main
parts/groups of topics:
PART I: Asymptotics.
This part includes methods for problems with
large (or small) parameters or variables (in ODEs and PDEs). A
tentative list of topics follows.
How are the zeros of the Riemann zeta function
related to Random Matrices and approach to equilibrium?
Prerequisites:
None. The course is largely self-contained. Some knowledge of
complex-variable theory and differential equations is assumed. (The
UMD courses MATH 414, MATH 462, or MATH 463, for example, would
suffice, but they are not required. Handouts with reviews will be
given in class whenever appropriate. If you are in any doubt about
your background, please consult with the instructor.
Class Times: Tuesday and Thursday: 11:00am - 12:15pm.
Location: MTH 0409
Handouts:
Many handouts that were used in the course in previous years can be
downloaded from the course webpage. Some of these handouts are
summaries of class lectures. Others are supplementary material.
These notes were written by Prof. Margetis and by students that were
taking the class. They are based on the books listed in the suggested bibliography. | 677.169 | 1 |
45
Algebra for non-specialists in secondary schools
This one-day course is designed for non-specialist subject teachers and HLTAs who have to teach maths and who wish to develop both their subject knowledge and classroom practice in algebra. The course is highly practical. It will look at making links between the different areas within the subject and the development of the subject based on the mastery approach to learning.
In-School Training
This course is available as in-school training. To view the in-school training details please click here
Aimed at
Non maths specialists and HLTAs who have to teach maths at secondary level.
Course details
This hands and minds-on course will develop participants' practical mathematical understanding and application within the classroom. It will improve their confidence in teaching algebra and enable them to effectively identify and address student's misconceptions. Participants will be better equipped to tackle the challenge that teaching the new maths curriculum presents.
By the end of the day, participants will have improved knowledge and understanding of how to teach:
Simple substitution
Solving linear equations from concrete methods
Factorising and its usefulness in calculation
Linking algebra to number and geometry
Pulling it all together and beginning to look at quadratic expressions
Session timetable
Session 1
Simple substitution
A mixture of tutor led, practical and discussion sessions
Session 2
Solving linear equations from concrete methods
A mixture of tutor led, practical and discussion sessions
Session 3
Factorising and its usefulness in calculation.
A mixture of tutor led, practical and discussion sessions
Session 4
Linking algebra to number and geometry.
Pulling it all together and starting to look at quadratic expressions
A mixture of tutor led, practical and discussion sessions
Tutor information
The tutor for improving progress and results in the most challenging of environments. He has built a repu-tation as being an excellent adviser, trainer and teacher. | 677.169 | 1 |
Algebra (I)
Understand that … 1. algebra involves using symbols to represent relationship 2. algebra is the link between abstract and concrete 3. arithmetic rules are adhered to in Algebra
Answer the following questions... 1. What is algebra? 2. What is the role of algebra? 3. How does algebra explain and predict relationships? 4. How can algebra translate the abstract to concrete and vice versa?
By the end of Chaper 4
You will be able to ... - use symbols to represent unknown in relationships. - express a relationship using symbols.
In the course of learning, we shall attempt to answer the following questions: 1. How can simple real world situations be transformed into algebraic terms? 2. What are the characteristics of the various forms of algebraic statements? 3. What inference can be made from any given algebraic statements? (expressions, equations, inequalities)? | 677.169 | 1 |
Lamp Modules Philosophy and Questions
Using computers and appropriate softward can provide highly effective tools
for promoting active learning, which leads to a deeper understanding of the
concepts. At the same time, this approach should help you develop
confidence in your ability to read, use, and then communicate
linear algebra to others.
Linear algebra has been designated a computer intensive course, and we
will be using Maple along with Lamp in order to satisfy this designator.
During some parts of the class, the class will meet in the computer
lab so that you can work through the Lamp modules on line.
Complete solutions for all exercises that appear in the modules are
provided in viewable sections so that you can get immediate feedback and
monitor your progress in understanding the material.
Dr. Sarah
will circulate to answer questions and engage in brief discussions.
For each Lamp demo that we complete, be
sure to answer the following questions on a piece of paper so that you
are prepared to share your answers with the rest of the class.
Is there anything that surprised you about this demo? If so
then explain.
Is there anything about this demo that confused you? If so then
note which part.
What are the most important examples in this demo? Next to each
such example (list the equations, matrix, or relevant details),
be sure to explain what they illustrated that you found
important or interesting. | 677.169 | 1 |
Now in its 8th edition, MATHEMATICS FOR PLUMBERS AND PIPEFITTERS delivers the essential math skills necessary in the plumbing and pipefitting professions. Starting with a thorough math review to ensure a solid foundation, the book progresses into specific on-the-job applications, such as pipe length calculations, sheet metal work, and the builder's level.Broad-based subjects like physics, volume, pressures, and capacities round out your knowledge, while a new chapter on the business of plumbing invites you to consider an exciting entrepreneurial venture. Written by a Master Plumber and experienced vocational educator, MATHEMATICS FOR PLUMBERS AND PIPEFITTERS, 8th Edition includes a multitude of real-world examples, reference tables, and formulas to help you build a rewarding career in the plumbing and pipefitting trade | 677.169 | 1 |
Math
Math Department
James Monroe High School
Mathematics
Course Descriptions
Middle School: The Mathematics courses for students in grades 7 and
8 at James Monroe High
School are Math 7 and Math 8. Each of these foundational courses uses the
Common Core Learning Standards to develop students' problem solving and
reasoning skills. These courses provide
the fundamentals of mathematics, therefore providing a solid foundation for
students to build on once they get to high school. So much of the high school mathematics
curriculum is based off the prior knowledge that students acquire in middle
school. Without a solid foundation of
knowledge, high school mathematics becomes even more challenging for our
students.
High School: The Mathematics courses for students in grades 9
– 12 at James Monroe High School are Algebra I, Geometry R, Algebra II, Pre-calculus,
and AP Statistics. In addition to these
traditional math courses, here at James Monroe High School exclusively, we have
the privilege of offering a College and Career Ready Math Course. Since mathematics tends to be an area of
struggle for students, the hope is that they will be better prepared for any
entry-level college math course. By
providing a wide range of upper-level math courses, students have choice in how
they further their knowledge of mathematics, whether it be exploring real-life
data and statistics or prepare themselves for college placement tests. With choice comes opportunity, and with
opportunity comes success | 677.169 | 1 |
Mathematics
Mathematics
Core
Specification
GCSE Mathematics
Course Structure
The GCSE Mathematics course has a two-tier structure that gives all students the opportunity of achievement. The higher tier course assesses work from grades 4 to 9 and the foundation tier course assesses work of grades 1 to 5.
Assessment
100% External Examination (three written papers, two with a calculator, one without)
Additional opportunities
The Mathematics curriculum offers a comprehensive range of experiences, activities, challenge and learning opportunities to meet the needs of individual students.
Skills Developed
You will develop a range of useful skills including logical thought, analytical techniques, accuracy, reasoning skills and the ability to think in abstract ways.
Progression Routes
GCSE Mathematics is an essential qualification for many jobs and is needed to meet entry requirements to access to Further and Higher education courses | 677.169 | 1 |
9780072424287College Algebra: Graphs and Models
Mathematical reform is the driving force behind the organization and development of this new college algebra text. The use of technology, primarily graphing utilities, is assumed throughout the text. The development of each topic proceeds from the concrete to the abstract and takes full advantage of technology, wherever appropriate. The first major objective of this book is to encourage students to investigate mathematical ideas and processes graphically and numerically, as well as algebraically. Proceeding in this way, students gain a broader, deeper, and more useful understanding of a concept or process. Even though concept development and technology are emphasized, manipulative skills are not ignored, and plenty of opportunities to practice basic skills are present. A brief look at the table of contents will reveal the importance of the function concept as a unifying theme. The second major objective of this book is the development of a library of elementary functions, including their important properties and uses. Having this library of elementary functions as a basic working tool in their mathematical tool boxes, students will be able to move into calculus with greater confidence and understanding. In addition, a concise review of basic algebraic concepts is included in Appendix A for easy reference, or systematic review. The third major objective of this book is to give the student substantial experience in solving and modeling real world problems. Enough applications are included to convince even the most skeptical student that mathematics is really useful. Most of the applications are simplified versions of actual real-world problems taken from professional journals and professional books. No specialized experience is required to solve any of the applications | 677.169 | 1 |
Use linear systems and matrices to analyze such questions as these: How can the stopping distance of a car be estimated based on three data points? How does computer graphics perform transformations and rotations? How can traffic flow along a network of roads be modeled? | 677.169 | 1 |
MIT Linear Finite Element Analysis
Instructor: Klaus-Jürgen BatheThis video series is a comprehensive cour
Help a friend. Share now on
Course Created by Massachusetts Institute of TechnologyThis video series is a comprehensive course of study that presents effective finite element procedures for the linear analysis of solids and structures. The finite element method is the ideal tool for solving static and dynamic problems in engineering and the sciences. Linear analysis assumes linear elastic behavior and infinitesimally small displacements and strains. To establish appropriate models for analysis, it is necessary to become familiar with the finite element methods available. | 677.169 | 1 |
19 Nov 2015
views:139515
TabletClass Math learn the basics of calculus quickly. This video is designed to introduce calculus concepts for all math students and make the topic easy to understand.21 Aug 2013
views:13999914 Mar 2013
views:241243
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
published:08 Jul 2013
views:405222 function and is Euler's number. First, we may demonstrate that the derivative of is .26
Calculus 1 Lecture 1.1: An Introduction to Limits
Calculus 1 Lecture 1.1: An Introduction to Limits
Calculus 1 Lecture 1.1: An Introduction to Limits
What is Cal21:58
Understand Calculus in 10 Minutes
Understand Calculus in 10 Minutes
Understand Calculus in 10 Minutes
TabletClass Math learn the basics of calculus quickly. This video is designed to introduce calculus concepts for all math students and make the topic easy to understand.What is calculus? (KristaKingMath)Calculus: What Is It?Calculus at a Fifth Grade Level intelligent1:32
What is Calculus?
What is Calculus?
What is Calculus?
This clip provides an introduction to Calculus. More information can be found at
15:37
❤︎² Basic Integration... How? (mathbff)
❤︎² Basic Integration... How? (mathbff)5:03
Meet 2 students who earned perfect score on AP calculus exam
Meet 2 students who earned perfect score on AP calculus exam p...
published: 28 Apr 2017
Calculus 1 Lecture 1.1: An Introduction to LimitsW... 19 Nov 2015
Understand Calculus in 10 Minutes
TabletClass Math learn the basics of calculus quickly. This video is designed to introduce calculus concepts for all math students and make the topic easy to understand.
Calculus I in 20 Minutes (The Original) by Thinkwell 21 Aug 2013 calculus...
published: 14 Mar 2013published: 08 Jul 2013published: 09 May 2017
1. What is Calculus | (Hindi)
why study differentiation and integration
published: 22 Jan 2016
Introduction to Calculus (1 of 2: Seeing the big picture) ...
published: 03 Oct 2007 e... I - Lecture 01 - A Review of Pre-Calculus m...
published: 19 May 2011
What is Calculus?
This clip provides an introduction to Calculus. More information can be found at
published: 17 Feb 2009
published: 01 May 2016 geom... inv... pre......What is calculus? (KristaKingMath)
► My ebook:
"What is calculus?" is a question many calculus students never learn the answer to!
Understanding wh...Calculus: What Is It?
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wonderi...
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
Calculus at a Fifth Grade Level
The foreign concepts of calculus often make it hard to jump right into learning it. If you ever wanted to dive into the world of mathematics - or if you are jus... to Calculus.
Watch the next lessonWhat is Calculus Used For? | Jeff Heys | TEDxBozeman
This talk describes the motivation for developing mathematical models, including models that are developed to avoid ethically difficult experiments. Three diff... ... Khan...Meet 2 students who earned perfect score on AP calculus exam
In this edition of "CBS This Morning's" Pushing the Limits series, we met two high school students who not only conquered calculus, but also pulled off an achie...
In this edition of "CBS This Morning's" Pushing the Limits series, we met two high school students who not only conquered calculus, but also pulled off an achievement that can stump college professors. Of more than 302,000 students around the world who took the Advanced Placement calculus test last year, Landon Labuske and Cedrick Argueta were two of only 12 who achieved a perfect score. Chip Reid reports.► My ebook:
"What is calculus?" is a questi...This video shows how calculus is both interesting and useful. Its history, practical uses,...The foreign concepts of calculus often make it hard to jump right into learning it. If you...Burden Of Grief
The war is over The last battles are gone Swords laying broken My bloodwork is all done I sit down for calming My breath is lessening I�m starting to tremble My sight is clearing My head is weary A dreadful awakening What has driven me Into insanity Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger back on the shelf I look around As I raise from my rest Discover what I�ve done No life I have left My heart is in pieces My soul is laying bare Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegyingEAH, ABOUT THAT. Democrats chose to reopen the government after an assurance from the Senate majority leader. That plan's already in trouble. 01.22.18 9.53 PM ET ... "The reality is ... Within a matter of minutes on Monday, conservative HouseRepublicans were throwing cold water on whether a Senate commitment to holding an open-ended immigration reform process changes their calculus ... ........................... Read more ... ....
Just as with coffee, a perfect cup of tea can be fleeting and ephemeral. Someone makes one for you, somewhere — a mug of milky black tea, a fragrant cup of jasmine green — and if you catch the bug, you might spend endless mornings trying to replicate it with different tea bags, leaves, vessels and steeping times. One factor sometimes gets overlooked in that calculus. The heat of the water you use ... The difference will likely be striking....
Ed MorrisseyPosted at 12.01 pm on January 23, 2018 Share on Facebook. Share on Twitter. After three days of a stunt shutdown, Democrats are still left to answer this question — and with a lot less leverage than they had before ... — CNNPolitics (@CNNPolitics) January 23, 2018 ... What makes Cuomo's hostility to the concept of political trade-offs even stranger is that this isn't a new calculus, even in terms of DACA ...Free agency may change the calculus, but the Bears have glaring needs at wide receiver, outside linebacker, cornerback and offensive line — in that order ... ....
Religion emerged from a homogenous source as human cosmology expressed belief in the existence of the Omnipotent Being ... In calculus, the holy books and instruments of worship bought by Africans who from time immemorial were converted to borrowed religions ran into trillions of dollars that could develop Africa better than anywhere else on the globe ... All these naïve behaviour have put anxieties on our finances, making us more desolate ... ....
WASHINGTON • Illinois Sen. Tammy Duckworth was one of 28 Democratic senators who changed their votes Monday and approved a short-term budget extension, ending a government shutdown that had lasted through the weekend ... Meanwhile, her colleague, Sen ... Sen ... But as Trump cut off negotiations with Schumer while hammering him on Twitter for causing the shutdown, the middle-out solution pushed that election-year calculus aside ... Sens ... ....
VICE. Hi. You're promoting a film ... I mean Rex Tillerson, Trump's secretary of state, has said some things, but you know it's so hard to build diplomatic coalitions, to put pressure on people to change their calculus, it requires more than a statement, it requires working the phone, having a summit, pulling people together trying to figure out incentives and disincentives for the people who might be abetting the Burmese government....
A long, healthy life is a blessing. The good news. Americans are living longer than ever before. Life expectancy has increased dramatically in the last few decades as a result of major advances in science, technology, and medicine. An individual's life expectancy is his or her median lifespan ... Even better news ... More money is needed to sustain a longer lifespan, and this changes the calculus of Americans' retirement decisions ... ....
Economists widely expect the Federal Reserve to raise its benchmark interest rate three times in 2018 ...Lately, the U.S. economy has been on a tear — it appears growth averaged 3% over the last three quarters and stock prices are booming ... It really doesn't matter what the Fed does, as long as it doesn't throw the economy into recession—I hope that counts as a bad thing in the calculus of Fed hawks—because housing prices should increase....
Democrats Kristopher Larsen and Mark Williams are gathering signatures to run in the 2nd Congressional District primary election, meaning former University of ColoradoRegent Joe Neguse is more than likely going to have some company on the June primary ballot ...Rep ... "I really do think, at least during the primaries, the independents are going to be a major force and really change the calculus of the election." ... Rep ... .... | 677.169 | 1 |
Undergraduate Problem Solving Contest
Rules
You must be an undergraduate enrolled in coursework at the University of Utah (students
in the HSUP program are also eligible).
You must work independently.
Write your solutions clearly and show all of your work.
If there is more than one sheet, staple the pages together. Include your name, student
ID number, and email address.
Bring your solution to the T. Benny Rushing Tutoring Center by the deadline and have
the tutor at the desk mark the date and time on your solution. Drop it in the box
provided (behind the tutoring desk).
A winner will be decided on the basis of the best solution submitted. If no best solution
can be determined (i.e. there exist relatively identical solutions), the winner will
be the student who submitted the first best solution.
Each submission will be given 3 points for a fully correct solution and 1 point for
a partially correct solution. The winner of each problem will get a bonus of e points.
This is not exactly a rule as it is unenforceable, but do not try to find the solution
to a problem by looking it up in a book or by searching online. That goes against
the spirit of the contest.
Other Information
The overall winner of the annual contest will receive an all-expense-paid trip to
MathFest, the annual meeting of the Mathematical Association of America. While attending
the meeting, the winner will be able to participate in a national problem solving
contest. | 677.169 | 1 |
These six standards deal with functions, covering skills including; constructing and comparing linear, quadratic and exponential models, solving problems involving linear, quadratic and exponential situations, and interpreting expressions for functions in the terms of the situation they model.
These concepts may be taught at different times throughout your course, depending on the curriculum established in your district. TheseThis resource is just one of several resources in my 'store' that addresses the CCSS standards (I know I said standards standards, but CCSstandards looks awkward).
Check out the others for a complete set of all the AES resources, including | 677.169 | 1 |
In this course we will assign outside videos
for you to watch to introduce many of the topics. We will then go over the
topics in class and work some examples to see if you understand. We want the
class to be more interactive, and are also adding this recitation aspect to the
course to cover all your learning needs.
We will assign homework problems in the online
texts and other texts by giving you handouts. We will also provide you with the
answers and go over solutions in class, as needed (the recitation aspect). Many
of the problems will come either from, Linear Algebra and its Applications by
Gilbert Strang, as well as Matrix Analysis and Applied Linear Algebra by
Carl D. Meyer. These are both very good textbooks for certain aspects of the
subject, and if you can find a copy of either one online or at a bookseller, we
would recommend getting one or both, if it is not cost prohibitive. We would
recommend the Meyer book first. Another useful textbook is: Elementary Linear
Algebra by Anton and Rorres.
In addition to the homework that is not collected, but used to
prepare you for the class, we will also assign several application-based problem
sets, to show you how linear algebra is used. Questions like these will not be
on the exams or the final, but are used to give you exposure to linear algebra
beyond classroom exercises. You can work together on groups, but members of a
group may be called upon to explain your solutions. | 677.169 | 1 |
California Standards for 8-12:)
(:div1 style="margin-left:5px" :)
[table border=1 width=98% cellpadding=3]
[row]
[]Pre-Algebra
>>&<<
[row]
[](:showhide init=hide div=div81 lshow='+' lhide='-':) %exa c12%Algebra 1%% (see also: [[Standards/Is1al|International Standards for Algebra 1]])
(:div81 id=div81Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.
* [[ca1al1|%exa%1ALG.1.0%%]] Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
** %pra%1ALG.1.1%% Students use properties of numbers to demonstrate whether assertions are true or false.
* [[ca1al2|%exa%1ALG.2.0%%]] Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
* [[ca1al3|%exa%1ALG.3.0%%]] Students solve equations and inequalities involving absolute values.
* [[ca1al4|%exa%1ALG.4.0%%]] Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.
* [[ca1al5|%exa%1ALG.5.0%%]] Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
* [[ca1al6|%exa%1ALG.6.0%%]] Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
* |[[ca1al7|%exa%1ALG.7.0%%]] Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.
* [[ca1al8|%exa%1ALG.8.0%%]] Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
* [[ca1al9|%exa%1ALG.9.0%%]] Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
* [[ca1al10|%exa%1ALG.10.0%%]] Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
* [[ca1al11|%exa%1ALG.11.0%%]] Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
* [[ca1al12|%exa%1ALG.12.0%%]] Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms
* [[ca1al13|%exa%1ALG.13.0%%]] Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
* [[ca1al14|%exa%1ALG.14.0%%]] Students solve a quadratic equation by factoring or completing the square.
* [[ca1al15|%exa%1ALG.15.0%%]] Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
* [[ca1al16|%exa%1ALG.16.0%%]] Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
* [[ca1al17|%exa%1ALG.17.0%%]] Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
* [[ca1al18|%exa%1ALG.18.0%%]] Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.
* [[ca1al19|%exa%1ALG.19.0%%]] Students know the quadratic formula and are familiar with its proof by completing the square.
* [[ca1al20|%exa%1ALG.20.0%%]] Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
* [[ca1al21|%exa%1ALG.21.0%%]] Students graph quadratic functions and know that their roots are the x- intercepts.
* [[ca1al22|%exa%1ALG.22.0%%]] Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
* [[ca1al23|%exa%1ALG.23.0%%]] Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
* [[ca1al24|%exa%1ALG.24.0%%]] Students use and know simple aspects of a logical argument:
** %pra%1ALG.24.1%% Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
** %pra%1ALG.24.2%% Students identify the hypothesis and conclusion in logical deduction.
** %pra%1ALG.24.3%% Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
* [[ca1al25|%exa%1ALG.25.0%%]]Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:
** %pra%1ALG.25.1%% Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
** %pra%1ALG.25.2%% Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.
** %pra%1ALG.25.3%% Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.
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[](:showhide init=hide div=div82 lshow='+' lhide='-':) %exa c12%Geometry%%
(:div82 id=div82 geometry skills and concepts developed in this discipline are useful to all students. Aside from learning these skills and concepts, students will develop their ability to construct formal, logical arguments and proofs in geometric settings and problems.
*[[CA_GE-1]] Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
*[[CA_GE-2]] Students write geometric proofs, including proofs by contradiction.
*[[CA_GE-3]] Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.
*[[CA_GE-4]] Students prove basic theorems involving congruence and similarity.
*[[CA_GE-5]] Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
*[[CA_GE-6]] Students know and are able to use the triangle inequality theorem.
*[[CA_GE-7]] Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
*[[CA_GE-8]] Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
*[[CA_GE-9]] Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.
*[[CA_GE-10]] Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
*[[CA_GE-11]] Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
*[[CA_GE-12]] Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
*[[CA_GE-13]] Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
*[[CA_GE-14]] Students prove the Pythagorean theorem.
*[[CA_GE-15]] Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
*[[CA_GE-16]] Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.
*[[CA_GE-17]] Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
*[[CA_GE-18]] Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, {$ tan( x ) = \frac{sin( x )}{cos( x )} $}, {$ (sin( x ))^ 2 + (cos( x ))^2 = 1 $}.
*[[CA_GE-19]] Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
*[[CA_GE-20]] Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.
*[[CA_GE-21]] Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
*[[CA_GE-22]] Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
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[](:showhide init=hide div=div83 lshow='+' lhide='-':) %exa c12%Algebra 2%%
(:div83 id=div83 complements and expands the mathematical content and concepts of Algebra I and Geometry. Students who master Algebra II will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic and exponential functions, the binomial theorem, and the complex number system.
* [[ca2al1|%exa%2ALG.1.0%%]] Students solve equations and inequalities involving absolute value.
* [[ca2al2|%exa%2ALG.2.0%%]] Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
* [[ca2al3|%exa%2ALG.3.0%%]] Students are adept at operations on polynomials, including long division.
* [[ca2al4|%exa%2ALG.4.0%%]] Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
* [[ca2al5|%exa%2ALG.5.0%%]] Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.
* [[ca2al6|%exa%2ALG.6.0%%]] Students add, subtract, multiply, and divide complex numbers.
* [[ca2al7|%exa%2ALG.7.0%%]] Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.
* [[ca2al8|%exa%2ALG.8.0%%]] Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.
* [[ca2al9|%exa%2ALG.9.0%%]] Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation {$ y = a(x-b)^2 + c $}.
* [[ca2al10|%exa%2ALG.10.0%%]] Students graph quadratic functions and determine the maxima, minima, and zeros of the function.
* [[ca2al11|%exa%2ALG.11.0%%]] Students prove simple laws of logarithms.
**11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
**11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
* [[ca2al12|%exa%2ALG.12.0%%]] Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.
* [[ca2al13|%exa%2ALG.13.0%%]] Students use the definition of logarithms to translate between logarithms in any base.
* [[ca2al14|%exa%2ALG.14.0%%]] Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
* [[ca2al15|%exa%2ALG.15.0%%]] Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.
* [[ca2al16|%exa%2ALG.16.0%%]] Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.
* [[ca2al17|%exa%2ALG.17.0%%]] Given a quadratic equation of the form {$ ax^2 + by^2 + cx + dy + e = 0 $}, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.
* [[ca2al18|%exa%2ALG.18.0%%]] Students use fundamental counting principles to compute combinations and permutations.
* [[ca2al19|%exa%2ALG.19.0%%]] Students use combinations and permutations to compute probabilities.
* [[ca2al20|%exa%2ALG.20.0%%]] Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.
* [[ca2al21|%exa%2ALG.21.0%%]] Students apply the method of mathematical induction to prove general statements about the positive integers.
* [[ca2al22|%exa%2ALG.22.0%%]] Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.
* [[ca2al23|%exa%2ALG.23.0%%]] Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.
* [[ca2al24|%exa%2ALG.24.0%%]] Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.
* [[ca2al25|%exa%2ALG.25.0%%]] Students use properties from number systems to justify steps in combining and simplifying functions.
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[](:showhide init=hide div=div84 lshow='+' lhide='-':) %exa c12%Trigonometry%%
(:div84 id=div84 Trigonometry uses the techniques that students have previously learned from the study of algebra and geometry. The trigonometric functions studied are defined geometrically rather than in terms of algebraic equations. Facility with these functions as well as the ability to prove basic identities regarding them is especially important for students intending to study calculus, more advanced mathematics, physics and other sciences, and engineering in college.
*[[CA_TR-1]] Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians.
*[[CA_TR-2]] Students know the definition of sine and cosine as y-and x-coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions.
*[[CA_TR-3]] Students know the identity {$ cos^2(x) + sin^2(x) = 1 $}:
**3.1 Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).
**3.2 Students prove other trigonometric identities and simplify others by using the identity {$ cos^2(x) + sin^2(x) = 1 $}. For example, students use this identity to prove that {$ sec^2(x) = tan^2(x) + 1 $}.
*[[CA_TR-4]] Students graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift.
*[[CA_TR-5]] Students know the definitions of the tangent and cotangent functions and can graph them.
*[[CA_TR-6]] Students know the definitions of the secant and cosecant functions and can graph them.
*[[CA_TR-7]] Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.
*[[CA_TR-8]] Students know the definitions of the inverse trigonometric functions and can graph the functions.
*[[CA_TR-9]] Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.
*[[CA_TR-10]] Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities.
*[[CA_TR-11]] Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities.
*[[CA_TR-12]] Students use trigonometry to determine unknown sides or angles in right triangles.
*[[CA_TR-13]] Students know the law of sines and the law of cosines and apply those laws to solve problems.
*[[CA_TR-14]] Students determine the area of a triangle, given one angle and the two adjacent sides.
*[[CA_TR-15]] Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa.
*[[CA_TR-16]] Students represent equations given in rectangular coordinates in terms of polar coordinates.
*[[CA_TR-17]] Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form.
*[[CA_TR-18]] Students know DeMoivre's theorem and can give nth roots of a complex number given in polar form.
*[[CA_TR-19]] Students are adept at using trigonometry in a variety of applications and word problems.
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[]Mathematical Analysis
This discipline combines many of the trigonometric, geometric, and algebraic techniques needed to prepare students for the study of calculus and strengthens their conceptual understanding of problems and mathematical reasoning in solving problems. These standards take a functional point of view toward those topics. The most significant new concept is that of limits. Mathematical analysis is often combined with a course in trigonometry or perhaps with one in linear algebra to make a yearlong precalculus course.
*1.0 Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically.
*2.0 Students are adept at the arithmetic of complex numbers. They can use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. They know the proof of DeMoivre's theorem.
*3.0 Students can give proofs of various formulas by using the technique of mathematical induction.
*4.0 Students know the statement of, and can apply, the fundamental theorem of algebra.
*5.0 Students are familiar with conic sections, both analytically and geometrically:
**5.1 Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).
**5.2 Students can take a geometric description of a conic section—for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6—and derive a quadratic equation representing it.
*6.0 Students find the roots and poles of a rational function and can graph the function and locate its asymptotes.
*7.0 Students demonstrate an understanding of functions and equations defined parametrically and can graph them.
*8.0 Students are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity. They determine whether certain sequences converge or diverge.
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[](:showhide init=hide div=div85 lshow='+' lhide='-':) %exa c12%Probability and Statistics%%
(:div85 id=div85 an introduction to the study of probability, interpretation of data, and fundamental statistical problem solving. Mastery of this academic content will provide students with a solid foundation in probability and facility in processing statistical information.
*[[CA_PS-1]] Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.
*[[CA_PS-2]] Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
*[[CA_PS-3]] Students demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses.
*[[CA_PS-4]] Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families.
*[[CA_PS-5]] Students determine the mean and the standard deviation of a normally distributed random variable.
*[[CA_PS-6]] Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.
*[[CA_PS-7]] Students compute the variance and the standard deviation of a distribution of data.
*[[CA_PS-8]] Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
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[](:showhide init=hide div=div86 lshow='+' lhide='-':) %exa c12%Linear Algebra %% (see also: [[Standards/IsMx|International Standards for Matrix Algebra]])
(:div86 id=div86 general goal in this discipline is for students to learn the techniques of matrix manipulation so that they can solve systems of linear equations in any number of variables. Linear algebra is most often combined with another subject, such as trigonometry, mathematical analysis, or precalculus.
*1.0 Students solve linear equations in any number of variables by using Gauss-Jordan elimination.
*2.0 Students interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.
*3.0 Students reduce rectangular matrices to row echelon form.
*4.0 Students perform addition on matrices and vectors.
*5.0 Students perform matrix multiplication and multiply vectors by matrices and by scalars.
*6.0 Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.
*7.0 Students demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space.
*8.0 Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted
as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.
*9.0 Students demonstrate an understanding of the notion of the inverse to a square matrix and apply that concept to solve systems of linear equations.
*10.0 Students compute the determinants of 2 × 2 and 3 × 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds
spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.
*11.0 Students know that a square matrix is invertible if, and only if, its determinant is nonzero. They can compute the inverse to 2 × 2 and 3 × 3 matrices using row reduction methods or Cramer's rule.
*12.0 Students compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.
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[](:showhide init=show div=div997 lshow='+' lhide='-':) %exa c12%Advanced Placement%%
(:div997 id=div997 border='1px solid #998' padding=5px bgcolor=#fed :)
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[](:showhide init=hide div=div9971 lshow='+' lhide='-':) %exa c12%AP Probability and Statistics%%
(:div9971 id=div9971 a technical and in-depth extension of probability and statistics. In particular, mastery of academic content for advanced placement gives students the background to succeed in the Advanced Placement examination in the subject.
*1.0 Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events.
*2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
*3.0 Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.
*4.0 Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.
*5.0 Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.
*6.0 Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable.
*7.0 Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to thoe families.
*8.0 Students determine the mean and the standard deviation of a normally distributed random variable.
*9.0 Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.
*10.0 Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.
*11.0 Students compute the variance and the standard deviation of a distribution of data.
*12.0 Students find the line of best fit to a given distribution of data by using least squares regression.
*13.0 Students know what the correlation coefficient of two variables means and are familiar
with the coefficient's properties.
*14.0 Students organize and describe distributions of data by using a number of different
methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
*15.0 Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.
*16.0 Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution.
*17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.
*18.0 Students determine the P-value for a statistic for a simple random sample from a normal distribution.
*19.0 Students are familiar with the chi-square distribution and chi-square test and understand their uses.
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[](:showhide init=hide div=div9972 lshow='+' lhide='-':) %exa c12%AP Calculus%%
(:div99721 id=div9972When taught in high school, calculus should be presented with the same level of depth and rigor as are entry-level college and university calculus courses. These standards outline a complete college curriculum in one variable calculus. Many high school programs may have insufficient time to cover all of the following content in a typical academic year. For example, some districts may treat differential equations lightly and spend substantial time on infinite sequences and series. Others may do the opposite. Consideration of the College Board syllabi for the Calculus AB and Calculus BC sections of the Advanced Placement Examination in Mathematics may be helpful in making curricular decisions. Calculus is a widely applied area of mathematics
and involves a beautiful intrinsic theory. Students mastering this content will be exposed to both aspects of the subject.
* [[CA_Cal-1]] 1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:
**1.1 Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
**1.2 Students use graphical calculators to verify and estimate limits.
**1.3 Students prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.
* [[CA_Cal-2]] 2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
* [[CA_Cal-3]] 3.0 Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.
* [[CA_Cal-14]] 4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability:
**4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
**4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.
**4.3 Students understand the relation between differentiability and continuity.
**4.4 Students derive derivative formulas and use them to find the derivatives of algebraic,
trigonometric, inverse trigonometric, exponential, and logarithmic functions.
* [[CA_Cal-5]] 5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
* [[CA_Cal-6]] 6.0 Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
* [[CA_Cal-7]] 7.0 Students compute derivatives of higher orders.
* [[CA_Cal-8]] 8.0 Students know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule.
* [[CA_Cal-9]] 9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.
* [[CA_Cal-10]] 10.0 Students know Newton's method for approximating the zeros of a function.
* [[CA_Cal-11]] 11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.
* [[CA_Cal-12]] 12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.
* [[CA_Cal-13]] 13.0 Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.
* [[CA_Cal-14]] 14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
* [[CA_Cal-15]] 15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.
* [[CA_Cal-16]] 16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.
* [[CA_Cal-17]] 17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.
* [[CA_Cal-18]] 18.0 Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
* [[CA_Cal-19]] 19.0 Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square.
* [[CA_Cal-20]] 20.0 Students compute the integrals of trigonometric functions by using the techniques noted above.
* [[CA_Cal-21]] 21.0 Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators or computers or both to approximate integrals numerically.
* [[CA_Cal-22]] 22.0 Students understand improper integrals as limits of definite integrals.
* [[CA_Cal-23]] 23.0 Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.
* [[CA_Cal-24]] 24.0 Students understand and can compute the radius (interval) of the convergence of power series.
* [[CA_Cal-25]] 25.0 Students differentiate and integrate the terms of a power series in order to form new series from known ones.
* [[CA_Cal-26]] 26.0 Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
* [[CA_Cal-27]] 27.0 Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.
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Online Graphing Calculators
Online Help
The following link provides help with each section. It provides video tutorials through the examples in the book. Homework Help: You will need to type in the KEYWORD which is MB7 and the section number sample: 1.KEYWORD:MB7 1-1 Parent Resources:You will need to type in the KEYWORD which is MB7 and the word Parent sample: 1.KEYWORD:MB7parent go.hrw.com
Expectations Expectation Sheet BE ON TIME Keep your cell phone & music device off and away Keep up with assignments No Late Assignment will be accepted Be Polite & Show Respect to everyone in the classroom
Participation Points Worth 10% of the overall semester grade.Students will be given a daily participation point based on taking notes and actively working on the task at hand. Student are expected to take notes during direct instruction, collaborative work and presentations. Students may keep notes in a composition book or designated area in a 3-ring binder. If a student is absent, that participation point may be given if the student promptly turns in evidence of note-taking and/or a written summary of the events from the day(s) missed. This summary and/or notes must include the date of the absence as well as a clear title that corresponds with the description listed on the assignment calendar. Each day absent requires its own distinct notes and/or summary. The time frame for making up the participation points is equal to the number of days absent.
Homefun/Class Work Worth 15% of the overall semester grade. Homefun is scored on a 5 point scale. Homefun is assigned on most days and can be found on the Chapter Calendar in the google drive folder. Class Work consists of collaborative activities and/or team quizzes.
Discovery Activities Worth 25% of the overall semester grade.Discovery Activities are projects that provide an extension into mathematical application. These projects are both individual and group. Students will be assigned up to two per grading period. The objective of the discovery activities is to have students complete work that requires higher level thinking which will compel students to creatively evaluate, research, study and even teach the content.
Chapter Tests Worth 35% of the overall semester grade. Chapter Tests are an individual assessment of the students comprehension level of the chapters concepts. Test are given at the end of each chapter. There are no retakes for the specific chapter test. For the first two grading periods for each semester students take a "Mini-Final" which covers the material given during that grading period (approximately 2 chapters worth of material). This exam counts as a test score, in addition may replace a low test score if a student scores higher on the Mini-Final than on one of the previous chapter exams covered within that grading period.
Absent Work If you are absent for:
Homefun/Class Work- You will need to make up all missed assignments. The make up for missed assignments must be made up in the amount of time frame Test- The test will be administered to you in class the day you return. Discovery Activities - They are due the day that you return from an excused absence. No late assignments will be accepted.
Tardy Policy
For any tardy you may be sent to the office to obtain a pass to class. This results in loss of even more class time. So Be on time! See the West Hills student handbook for information regarding attendance policies.
Cell Phone Policy
Are to remain on silent or off during class time and put away in your backpacks. Texting or having your phone out during class is NOT allowed. Consequences could result in confiscation of the phone until the end of that school day. Defiance in the no phone policy may result in a behavioral referral.
Electronic Music Devices
Need to put away upon entering the classroom and remain put away & off. I usually allow you to listen to your music AFTER the chapter test. If your music device is allowed, you must always have on headphones & the volume must be at a level that is not heard by anyone around you. You must always be able to hear your team member and me.
Academic Dishonesty
Academic integrity is essential to your academic success. Dishonesty on homework, quizzes, and /or exams will result in a score of a zero and disciplinary action.
Parent/Student Contact Information
The form below provides parents with a brief description of the course along with a section for your parent/guardian to provide contact information. Click here to fill out the online form. | 677.169 | 1 |
problem solving of mathematics
1.
INTRODUCING
Mathematics learning involves the acquisition of knowledge and skills
especially problem solving skills. In real life, problem solving becomes the focus
while knowledge is only the accessory. This is because, not a single day passes
without we having to solve problems. So, the need for the problem solving approach
in teaching mathematics.
WHAT IS A PROBLEM?
A problem is a statement or a situation where there is an obstacle between us
and what we want. Problems are generally classified as routine and non-routine.
WHAT IS PROBLEM SOLVING?
Problem solving is the ability to overcome or remove the obstacle so that we
can get what we want. Problem solving is a process. It requires critical thinking,
ability to make decisions, use the correct strategy to find the solution and check the
result.
MODEL FOR PROBLEM SOLVING
The most commonly used model is that of George Polya (1973), who proposed 4
stages in problem solving, namely :
1. Understand the problem
2. Devise a strategy for solving it
3. Carry out the strategy
4. Check the result
2.
QUESTION 1
In your words give the definition for :
a) Routine problem
b) Non-routine problem
QUESTION 2
Solve the following problems using Polya's model in 2 different strategies.
A police station has 25 vehicles consisting of motorcycles and cars. The total
number of tyres of both motorcycles and cars equal to 70. Find the number of
motorcycles and cars the station has.
QUESTION 3
A T-shirt costs RM 6.50. A pair of socks RM 0.85. Azmer bought three pairs of socks and a
T-shirt. How much did he pay?
Answer for question 1:
a) Routine problem is definied as a problem in mathematic lesson that involves
easy and simple problem solving. It present a question to be answered with
out need certain strategies. It means, the routine problem can be solved by
direct application of previously learned algorithms. Example:
Ali eat 2 piece of cakes. 5 minutes later, he eat 1 more piece of cakes. How
many piece of cakes that Ali eat?
Solution: 2 piece of cakes + 1 pieces of cakes = 3 piece of cakes
3.
b) Non-routine is definied as a problem in mathematic lesson that involves
difficult problem solving. It means, solving the non-routine problem need us to
think analytically based on the problem. It requires us to use our cognitive by
using the critical and creative thinking skills. It also need a solution in which
applying the skills, acquired knowledge and understanding to a new and
unfamiliar situations in order to solve it.
Answer for question 2
1. For this question, the first strategy that we use is the Drawing or Sketches's
strategy.
Step 1: FIND OUT
First, we must draw the vehicles with two tyres. Then, we must add the tyres
until the number of tyres equal to 70. After that, we can see how much
motorcycles and cars.
Step 2: CHOOSE A STRATEGY
How should we approach this problem? We can make skatches.
Step 3: SOLVE IT
Before we add 2 more tyres to make the number of tyres become 70:
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
4.
After we add 2 more tyres to make the number of tyres become 70:
OOOO OOOO OOOO OOOO OOOO
OOOO OOOO OOOO OOOO OOOO
OO OO OO OO OO
OO OO OO OO OO
OO OO OO OO OO
Sign: OOOO – Car OO - Motorcycle
From this sketches, we can see how much number of motorcycles and cars at
the police station. There are 15 motorcycles and 10 cars in the police
station.
Step 4: LOOK BACK
Did we answer the correct question, and does our answer seem reasonable?
Yes. (If you want to know our answer is correct or not, you must count the
number of vehicle's tyres).
2. The second strategy that we use is Make a Chart's strategy.
Step 1: FIND OUT
5.
What is the question we have to answer? How many motorcycles and cars in
the police station.
How many vehicles in the police station? 25 vehicles.
How many number of vehicle's tyres in police station? 70 tyres.
How many tyres that motorcycles have? 2 tyres.
How many tyres that cars have? 4 tyres.
Step 2: CHOOSE A STRATEGY
What strategy will help here? We could model this on paper, but accuracy
would suffer. We could also use equations. But, let's make a table.
Step 3: SOLVE IT
Firstly, we make a table with 5 rows and 3 columns. Then, we choose our
target. For example, in the police station have 6 cars and 19 motorcycles. So
we can see the total of vehicles in the police station is 62 vehicles. Then, we
try and error with the same ratio until we get the answer which are 15
motorcycles and 10 cars:
CARS
( 4 TYRES)
MOTORCYCLES
(2 TYRES)
TOTAL OF VEHICLES
(70 TYRES)
6 19 62
7 18 64
8 17 66
9 16 68
10 15 70
Calculation:
1 car have 4 tyres. 1 motorcycles have 2 tyres.
So, if there has 6 cars and 19 motorcycles...
6.
6 x 4 = 24 and 19 x 2 = 38
And the total of vehicles are 62 (24+38). But, the answer is wrong.
So, we try and error with the same ratio until we get the correct answer which
are 10 cars and 15 motocycles:
10 x 4 = 40 and 15 x 2 = 30. The total of vehicles are 70 (40 + 30). The
answer is correct.
Step 4: LOOK BACK
Did we answer the question asked? Yes.
Does our answer seem reasonable? Yes.
Answer for question 3:
Step 1
Understanding the problems
- Given : The price of a T-shirt = RM 6.50
- The price of pair of socks = RM 0.85
- Find the total costs of three pairs of socks and a T-shirt.
Step 2
Devising a plan
- Find the price of three pairs of socks by using addition and then add with a T-shirt.
Step 3
Carry out the plan
First strategy
7.
Price of three of socks
RM 0.85 + RM 0.85 + RM 0.85 = RM 2.55
Add with the price of a T-shirt = RM 2.55 + RM 6.50
= RM 9.05
Second strategy
Find the price of three pairs of socks by using multiplication and add with the price of the T-
shirt.
Price of three of socks
- RM 0.85 x 3 = RM 2.55
- Add with the price of a T-shirt = RM 2.55 + RM 6.50
= RM 9.05
Step 4
Looking back
Check: By using an addition, check the answer.
Three socks: RM 0.85 + RM 0.85 + RM 0.85 + RM 6.50
= RM 9.05
Suggested strategy
The most efficient strategy to get the answer of this question is the second strategy. The
second strategy is using multiplication strategy then adds with the other price.
8.
A T-shirt costs RM 6.50. A pair of socks is cost RM 0.85. Azmer bought three pairs of socks
and a T-shirt. How much did he pay?
- RM 0.85 x 3 = RM 2.55
- Add with the price of a T-shirt = RM 2.55 + RM 6.50
= RM 9.05
This strategy is efficient because we can find the answer quickly. The step is also easier and
no need to use many step. We just use to times the price of 3 pair of socks and then add
with the T-shirt price. | 677.169 | 1 |
LaTeX
Description: LaTeX is a macro package created to typeset documents attractively and consistently. This guide to the LaTeX markup language can serve as a useful resource for everyone from new users who wish to learn, to old hands who need a quick reference.
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Getting Started with LaTeX by David R. Wilkins - Trinity College, Dublin LaTeX is particularly suited to the production of long articles and books, since it has facilities for the automatic numbering of chapters, sections, theorems, equations etc., and also has facilities for cross-referencing. (3980 views) | 677.169 | 1 |
Most effective math problem solver That will Cause you to a far better Pupil
In certain classes, all it will require to move an exam is be aware taking, memorization, and recall. However, exceeding inside a math class takes another form of effort. You can not only present up for the lecture and observe your teacher "talk" about geometry and . You find out it by undertaking: paying attention in class, actively studying, and solving math problems – even when your instructor hasn't assigned you any. When you find yourself battling to complete very well in the math course, then take a look at best site for solving math problems to learn how you could become a much better math scholar.
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Math courses stick to a all-natural progression – each builds upon the awareness you've acquired and mastered in the previous system. In case you are discovering it difficult to observe new concepts at school, pull out your outdated math notes and assessment former substance to refresh yourself. Make sure that you meet the conditions prior to signing up for just a class.
Evaluation Notes The Night In advance of Class
Detest whenever a teacher phone calls on you and you have overlooked tips on how to solve a specific issue? Keep away from this moment by examining your math notes. This will enable you to determine which concepts or questions you'd want to go in excess of at school the following working day.
The considered undertaking research every night time could seem annoying, but when you'd like to reach , it really is essential that you consistently practice and master the problem-solving techniques. Make use of your textbook or on-line guides to work via prime math problems over a weekly basis – even if you have got no research assigned.
Utilize the Supplements That include Your Textbook
Textbook publishers have enriched modern-day publications with extra product (including CD-ROMs or on the net modules) that will be used to aid students obtain further apply in . A few of these components may incorporate an answer or rationalization guide, which could enable you to with functioning via math difficulties all on your own.
Go through Ahead To stay Ahead
If you'd like to minimize your in-class workload or maybe the time you shell out on homework, make use of your free time after university or within the weekends to read in advance to your chapters and ideas which will be included the following time you might be in school.
Critique Previous Tests and Classroom Examples
The function you are doing in class, for homework, and on quizzes can give clues to what your midterm or closing examination will look like. Make use of your previous assessments and classwork to produce a own analyze manual for your personal forthcoming exam. Search for the way your instructor frames thoughts – this is often possibly how they will show up on the test.
Learn to Perform Via the Clock
This can be a common examine idea for folks getting timed exams; specially standardized assessments. Should you only have forty minutes for just a 100-point test, you'll be able to optimally invest 4 minutes on each 10-point concern. Get details regarding how extensive the examination might be and which kinds of issues are going to be on it. Then prepare to attack the better issues first, leaving yourself plenty of the perfect time to spend within the far more complicated ones.
Improve your Resources to obtain math homework assistance
If you're acquiring a tough time knowledge principles in class, then make sure you get help beyond class. Talk to your pals to make a study group and visit your instructor's workplace hours to go more than challenging complications one-on-one. Attend analyze and review periods once your teacher announces them, or seek the services of a non-public tutor if you want a person.
Speak To You
When you are examining issues for an examination, check out to explain out loud what tactic and procedures you accustomed to get your methods. These verbal declarations will come in helpful during a examination after you really need to remember the techniques you should choose to find a answer. Get more practice by seeking this tactic using a mate.
Use Examine Guides For More Observe
Are your textbook or class notes not aiding you have an understanding of everything you should be discovering in school? Use analyze guides for standardized examinations, like the ACT, SAT, or DSST, to brush up on previous content, or . Analyze guides generally appear outfitted with extensive explanations of how to fix a sample trouble, , and you also can typically find where by may be the improved invest in mathtroubles. | 677.169 | 1 |
This expanded edition of the original bestseller, How to Teach Mathematics, offers hands-on guidance for teaching mathematics in the modern classroom setting. Twelve appendices have been added that are written by experts who have a wide range of opinions and viewpoints on the major teaching issues.
Eschewing generalities, the award-winning author and teacher, Steven Krantz, addresses issues such as preparation, presentation, discipline, and grading. He also emphasizes specifics--from how to deal with students who beg for extra points on an exam to mastering blackboard technique to how to use applications effectively. No other contemporary book addresses the principles of good teaching in such a comprehensive and cogent manner.
The broad appeal of this text makes it accessible to areas other than mathematics. The principles presented can apply to a variety of disciplines--from music to English to business. Lively and humorous, yet serious and sensible, this volume offers readers incisive information and practical applications.
"synopsis" may belong to another edition of this title.
About the Author:
Steven G. Krantz, Washington University, St. Louis, MO, USA.
Review:
"Since the first edition of How to Teach Mathematics the increasing maturity of both traditionalist and reform movements has given Krantz more insights into the teaching of mathematics. The book is intended primarily for the graduate student or novice instructor; however, the book is also valuable for others. Post-secondary instructors ... Mathematics department heads ... Teaching Development Centers ... university administrators. In the appendices twelve other mathematics teachers comment in some way on Krantz's text and give some insight into other approaches to teaching. This book is a must read for instructors preparing their courses for next semester." ---- MAA Online
"An original contribution to the educational literature on teaching mathematics at the post-secondary level. The book itself is an explicit proof of the author's claim `teaching can be rewarding, useful, and fun'." ---- Zentralblatt MATH
"Unlike secondary school teachers, college and university teachers usually have no preliminary theoretical background in the teaching of mathematics. [This book] is written in a lively and humorous style, even though the points discussed are entirely serious and sensible. The author succeeds in elucidating the fine points of excellent teaching and offers a lot of important practical advice. The book is strongly recommended to everybody who teaches mathematics." ---- European Mathematical Society Newsletter | 677.169 | 1 |
Limits And Derivatives Class 11 Notes
In mathematics, a limit can be defined as a values of the input function and some value and derivatives are used to measure the sensitivity of one with respect to another variable. Both Limits and derivatives serve as the entry point to calculus for class 11 students. Limits are mainly used to find the values of a function nearer to some values.
Limits and derivatives are mainly used for finding the slopes of any curves, the rate of heat transfer and another field of both science and in mathematics.
Limits and Derivatives Class 11 Notes available on our website Byju's .com is mainly for those student appearing for class 11 exams. These notes are well prepared based on the NCERT solutions along with the main points, necessary formulas revision notes and other important questions.
Check the link given below for more detailed information about Limits and Derivatives Class 11 Notes. | 677.169 | 1 |
Designed for the first-year developmental math course in beginning algebra, this text retains the hallmark features that have given the Aufmann texts a solid reputation for reliability: a clear writing style, an emphasis on problem-solving strategies, and the acclaimed Aufmann Interactive Method. The text's objective-based framework offers guided learning for both lecture and self-paced courses.
"synopsis" may belong to another edition of this title.
About the Author:
Richard Aufmann is the lead author of two bestselling developmental math series and a bestselling college algebra and trigonometry series, as well as several derivative math texts. He received a BA in mathematics from the University of California, Irvine, and an MA in mathematics from California State University, Long Beach. Mr. Aufmann taught math, computer science, and physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum, and the impact of technology on curriculum development. | 677.169 | 1 |
Mathematical Modeling and Computer Simulation / Edition 1
Daniel Maki and Maynard Thompson provide a conceptual framework for the process of building and using mathematical models, illustrating the uses of mathematical and computer models in a variety of situations. This text helps students learn that model building is a dynamic process involving simplification, approximation, abstraction, analysis, computation, and comparison. Students begin the process of model building with a consideration of phenomena arising in another academic area or in the real world.
Editorial Reviews
"The authors focus on the process of model building and the subsequent analysis and evaluation. They make a clear effort to get the student to think about the material. Also, the exercises are well thought out, and are not simply ones that students can do by mimicking examples provided in the text."
"I was very impressed with the overall approach to the topic of Mathematical Modeling. I felt that the introduction to the subject of Modeling was one of the best that I have ever read." | 677.169 | 1 |
Key To...Series
Key To Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language and examples are easy to follow; word problems relate algebra to familiar situations, helping students understand abstract concepts. Students will develop understanding by solving equations and inequalities intuitively before formal solutions are introduced.
Key To Decimals offers a unique, proven way to introduce decimals to your students. New concepts are explained in simple language and examples are easy to follow; word problems relate decimals to familiar situations, helping students understand abstract concepts. Students develop understanding by progressing slowly from basic to more complex questions.
Whether you're introducing fractions for the first time, or find that your students need review and additional practice, Key to Fractions covers all major topics. Written with secondary students in mind, these self-paced workbooks provide the reinforcement needed for fraction mastery. Book 1 teaches fraction concepts, Book 2 teaches multiplying and dividing, Book 3 teaches adding and subtracting, and Book 4 teaches mixed numbers. Each book has a practice test at the end.
Key to Percents
Key to Percents emphasizes mental computation and estimation skills—since most work with percents is done without pencil and paper—before moving on to solving problems using equal fractions and decimal multiplication. Word problems in a variety of applications are also included. This series assumes only knowledge of fraction and decimal computation.
Key to Geometry offers a non-intimidating way to prepare students for formal geometry as they do step-by-step constructions. Using only a pencil, compass, and straightedge, students begin by drawing lines, bisecting angles, and reproducing segments. Later they do sophisticated constructions involving over a dozen steps and are prompted to form their own generalizations.
Key to Measurements is an exceptional study series for those who need to perfect their understanding of measurements. Book 1 contains interesting and fun activities related to measuring length; Book 2 focuses on length, perimeter, and area measures; Book 3 develops further the concept of area and Book 4 covers a variety of measurement topics.
A fantastic aid to those who need to perfect their understanding in Metric Measurements! The only system used in international commerce and communication, and one becoming more and more prevalent in the U.S, it's becoming very important to have a firm grasp on the Metric system. | 677.169 | 1 |
Pages
Thursday, November 17, 2011
GeoGebra
GeoGebra is a free software program for mathematics instruction and would be a great resource for any algebra, geometry, statistics, or calculus classroom. The interface is divided into three sections, including an input bar for equations and functions, an algebra view to see and edit equations and functions, and a graphic view that will allow the teacher/student to construct objects and see the graphs of functions. The construction tools for this software are vast and include options such as: lines through points, bisectors, tangents, locus, intersections, polygons, compasses, circular arcs, ellipse, hyperbole, area, slope, and the list goes on. Teachers and students can use this software to draw graphs, transform graphs, construct parallelograms, find mean, median, and mode, construct a tangent, and much more. If needed, teachers and students can save a GeoGebra document as a ggb (GeoGebra file), gif, png, or your file can be printed. In addition, you can export as a "Dynamic Worksheet as Webpage (HTML)" to easily share the file by email or link to your website.
The GeoGebra software can be downloaded to your computer at the GeoGebra website or you can run a fully functional applet in your web browser. If you run the applet version, you will have to allow the Java plug-in to run on your computer. In addition, the GeoGebra website houses a Materials section, including interactive worksheets, collections, and tutorials.
Classroom Integration Resources:
The links below will provide resources to get you started as well as classroom lessons/resources | 677.169 | 1 |
General Algebra
The course will familiarise students with basics of modern algebra. We will describe general properties of universal algebras and study, in more detail, individual algebraic structures, i.e., groupoids, semigroups, monoids, groups, rings and fields. Particular emphasis will be placerd on groups, rings (especially the ring of polynomials) and finite (Galois) fields.
Students will be made familiar with the basics of general algebra. It will help them to realize numerous mathematical connections and therefore to understand different mathematical branches. The course will provide students also with useful tools for various applications.
Mode of delivery
90 % face-to-face, 10 % distance learning
Prerequisites
The students are supposed to be acquainted with the fundamentals of linear algebra taught in the first semester of the bachelor's study programme.
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures an on getting acquainted with algebraic software.
Assesment methods and criteria linked to learning outcomes
The course-unit credit is awarded on condition of having attended the seminars actively and passed a written test. The exam has a written and an oral part. The written part tests student's ability to deal with various problems using the knowledge and skills acquired in the course. In the oral part, the student has to prove that he or she has mastered the related theory.
Language of instruction
Czech
Work placements
Not applicable.
Aims
The aim of the course is to provide students with the fundamentals of modern algebra, i.e., with the usual algebraic structures and their properties. These structures often occur in various applications and it is therefore necessary for the students to have a good knowledge of them.
Specification of controlled education, way of implementation and compensation for absences
Since the attendance at seminars is required, it will be checked systematically by the teacher supervising the seminar. If a student misses a seminar, an excused absence can be compensated for via make-up topics of exercises.
Type of course unit
1. Operations and laws, the concept of a universal algebra
2. Some important types of algebras
3. Basics of the group theory
4. Subalgebras, decomposition of a group (by a subgroup)
5. Homomorphisms and isomorphisms
6. Congruences and quotient algebras
7. Congruences on groups and rings
8. Direct products of algebras
9. Ring of polynomials
10.Divisibility and integral domains
11.Gaussian and Euclidean rings
12.Mimimal fields, field extensions
13.Galois fields | 677.169 | 1 |
Basic Mathematics by M.E. Wardle (Paperback, 1990)
£1.86Designed specifically for GCSE levels 1 and 2, and Standard Grade Foundation and General Levels, this is a complete source of questions for revision and practice. It can be used as a supplementary source of exercises at any point during the course, or for revision in the final year. | 677.169 | 1 |
Algebra Sentence Examples
LINK / CITEADD TO FLASH CARDS
The earliest algebra consists in the solution of equationsPrinciples of ordinary algebra; B.
But when I took up algebra I had a harder time still.
In algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving algebraical equations of every degree.
From this formula we obtain by elementary algebra 1) !
+a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch.
When 0=4 it is clear that no form, whose partition contains a part 3, can be reduced; but every form, whose partition is composed of the parts 4 and 2, is by elementary algebra reducible by means of perpetuants of degree 2.
The braille worked well enough in the languages, but when it came to geometry and algebra, difficulties arose.
The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics.
When algebra had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known.
His standard work on algebra, written in Arabic, and other treatises of a similar character raised him at once to the foremost rank among the mathematicians of that age, and induced Sultgn Malik-Shgh to summon him in A.H.
Young, The Algebra of Invariants (Cambridge, 1903).
ALGEBRA (from the Arab.
The principal step in the modern development of algebra was the recognition of the meaning of negative quantities.
The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and geometrical measurement.
But, when I took up Algebra, I had a harder time still--I was terribly handicapped by my imperfect knowledge of the notation.
From February to July, 1898, Mr. Keith came out to Wrentham twice a week, and taught me algebra, geometry, Greek and LatinSpecial kinds of algebra; C. History.
But on the night before the algebra examination, while I was struggling over some very complicated examples, I could not tell the combinations of bracket, brace and radical.
He also studied the first six books of Euclid and some algebra, besides reading a considerable quantity of Hebrew and learning the Odes of Horace by heart.
- The first to publish anything on Diophantus in Europe was Rafael Bombelli, who embodied in his Algebra (1572) all the problems of Books I.
Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra (Cambridge, 1885).
At the age of eight he began Latin, Euclid, and algebra, and was appointed schoolmaster to the younger children of the family.
His principal writings are Micrographia (1664); Lectiones Cutlerianae (1674-1679); and Posthumous Works, containing a sketch of his "Philosophical Algebra," published by R.
After Maclaurin's death his account of Newton's philosophical discoveries was published by Patrick Murdoch, and also his algebra in 1748.
For Tartaglia's discovery of the solution of cubic equations, and his contests with Antonio Marie Floridas, see Algebra (History).
Each definition gives rise to a corresponding algebra of higher complex numbers.
For the subjects of this general heading see the articles ALGEBRA, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.
Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.
This form of algebra was extensively studied in ancient Egypt; but, in accordance with the practical tendency of the Egyptian mind, the study consisted largely in the treatment of particular cases, very few general rules being obtained.
For many centuries algebra was confined almost entirely to the solution of equations; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use of the minus sign.
The development of symbolic algebra by the use of general symbols to denote numbers is due to Franciscus Vieta (Francois Viete, 1540-1603).
De Traytorrens, went through the elements of algebra and geometry, and the three fi r st books of the Marquis de l'Hopital's Conic Sections.
Cantor's histories of mathematics, and more elaborate analyses are those of Nesselmann (Die Algebra der Griechen, Berlin, 1842) and G.
But he seems to have been well cared for, and he was at the age of fourteen sufficiently advanced "in algebra, geometry, astronomy, and even the higher mathematics," to calculate a solar eclipse within four seconds of accuracy.
The Leiden copy of ~Omar Khayygms work on algebra was noticed as far back as i 742 by Gerald Meerman in the preface to his Specimen calculi fluxionalis; further notices of the same work by Sdillot appeared in the Nouv.
The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.
This theorem is due to Cayley, and reference may be made to Salmon's Higher Algebra, 4th ed.
This led to the idea of algebra as generalized arithmetic.
They teach further the solution of problems leading to equations of the first and second degree, to determinate and indeterminate equations, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called res and its square census.
The subject-matter of algebra will be treated in the following article under three divisions: - A.
Algebra and geometry were the only studies that continued to defy my efforts to comprehend them.
Again in January 1757 he writes: " I began to study algebra under M.
It is true that I was familiar with all literary braille in common use in this country--English, American, and New York Point; but the various signs and symbols in geometry and algebra in the three systems are very different, and I had used only the English braille in my algebra.
However, the braille worked well enough in the languages; but when it came to Geometry and Algebra, it was different.
In pure algebra Descartes expounded and illustrated the general methods of solving equations up to those of the fourth degree (and believed that his method could go beyond), stated the law which connects the positive and negative roots of an equation with the changes of sign in the consecutive terms, and introduced the method of indeterminate coefficients for the solution of equations.'
The first, second and third sections of this publication comprise respectively the papers communicated by him to the Academies of Sciences of Turin, Berlin and Paris; the fourth includes his miscellaneous contributions to other scientific collections, together with his additions to Euler's Algebra, and his Lecons elementaires at the Ecole Normale in 1795.
Consequently, I did not do so well as I should have done, if Teacher had been allowed to read the Algebra and Geometry to me. | 677.169 | 1 |
Algebra 2 Lesson Plan In this lesson, the
students will use second order
determinants and Cramer's Rule
to solve two equations and two
unknowns. One could easily
modify it to demonstrate 3X4
matrix operations.
The students should have a solid understanding of what second order determinants and Cramer's Rule are before beginning this unit. | 677.169 | 1 |
Mathematical application in textiles
Mathematics is interlinked to each and every processes involved in any field. Some of the important general applications of mathematics in
textile are as follows,
Conversions from one unit to other as different
countries have different set of units and to convert to common unit. E.g.
denier, tex, count, etc.
To arrive at a relationship between two or more
variables so that by knowing one variable we can find the other. E.g.
Tpi-count-twist multiplier, stitch length-wales-spacing, etc.
By using the area, volume and density of the shapes,
cross section of the fiber, density, volume and geometry of the structure
can be analyzed.
Production, Efficiency, Cover factor, Speed of the
machines from gearing, weft preparation calculations in weaving, beam
requirement in warping and in so many other applications in various department.
In Computer Color Matching
The main aim of the computer color matching is not only to
obtain the desired shade but also to analyze the various possibilities to get
the shades at minimum cost.
Matrix: In this method the various dye compositions, their
intensity, proportion, concentration and cost are treated as variables in
matrix and solved by trisimulus method to get the required datas.
Factorials: Factorials are used to explore how many combinations
can match the shade, which of them are economical or how close they are
when viewed in different light (meteamerism).
Partial Differentiation: It is used to predict the accuracy of the color,
alignment of dyes, reflectance measurement, saturation limit, compatibility and
differences in the strength and tone of the dye used.
Conclusion
In textile from cotton to apparel manufacturing every process is carried out by calculations. In order to get the required quality and production mathematical knowledge is essential especially for the management peoples. Instead of
going for testing the samples for identifying many numbers of variables to
arrive at the result, mathematical conversions and formulas are used for easy
calculations and time saving. These mathematical formulas are mostly applied in
textile sampling and testing. In order to for a new process in the industry
apart from the regular process, mathematical applications are involved to
obtain the optimum standards and settings | 677.169 | 1 |
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With this fascinating volume, Keith Devlin proves that the guiding principles of some of the most mysterious mathematical topics can be made comprehensible. Writing with an elegant lucidity, Devlin sh...
MATHEMATICAL APPLICATIONS FOR THE MANAGEMENT, LIFE, AND SOCIAL SCIENCES, 11th Edition, features a concept-based approach, multiple presentation methods, and interesting and relevant applications that ... | 677.169 | 1 |
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