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The purpose of is to provide information about the textbook Graph Theory and Its Applications and to serve as a comprehensive graph theory resource for graph theoreticians and students. |
Mathematics
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Did you know...
Multiple mathematics professors have been awarded an Endowed Teaching Chair.
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Seminole State Advance is a new program to strengthen the math abilities of students seeking careers in science, engineering or mathematics.
Whatever your math level, from basic arithmetic to differential equations, Seminole State College offers math courses that can help you learn new ways to approach problem-solving.
Seminole State's math students enjoy easy access to computers in classrooms and the fully staffed Academic Success Center. Statistical Methods I (STA2023) is taught entirely in computer labs, so students can work closely with their professors.
Whether you have math anxiety, find math challenging or desire a quality foundation in math, Seminole State's diverse array of math courses ensure that every student will learn in classes that meet their skill level and pacing.
Many courses are taught with a choice of traditional, distance learning or computer-based instruction to best suit your preferred learning style.
Math Department Facts
Math professor Martha Goshaw was awarded a national Teaching Excellence Award in 2003 and 2012.
The Math Department hosts the Seminole County High School Math Contest in cooperation with the Seminole County Teachers of Mathematics, where high school students compete in calculus, pre-calculus, algebra, geometry and statistics. Also, there is a scholarship competition to win a 60 credit-hour scholarship to Seminole State.
Calculator workshops for graphing calculators are offered at the beginning of each fall and spring term.
Near the end of most terms, the Math Department offers review sessions for classes with departmental finals. |
What''s Calculus all about?
Does Calculus have any relevance to our daily lives? Or is it just another conglomeration of mathematical symbols hardly making sense to most? This is a presentation to remove the 'fear' of Calculus among students and introduce the subject to a anyone who is a total stranger to the subject but not a stranger to Mathematics. The presentation starts with the nature and scope of Calculus and the type of problems solved using Calculus. Mention is also made of the mathematicians who 'invented' the subject. Some interesting curves are also shown in the end. |
what is pre-algebra?
At least, he's taking the same course I took in 9th grade lo these many years ago. IIRC
Ed seems to remember it all better than I do, and he says the same thing. Christopher is studying the stuff he studied as a freshman in high school.
What is pre-algebra, anyway?
Do we know?
There wasn't any pre-algebra when I was a kid, and I remember Carolyn expressing skepticism about the whole concept back when we first met.
These days I think pre-algebra is simply Year One in a 3-year Algebra Spiral. You teach Algebra 1 in 6th grade, calling it Pre-Algebra; then you teach Algebra 1 again in 7th grade; then you teach it again in 8th.
That's the fast track.
For the slow track you start teaching Algebra 1 in 7th or 8th, calling it Pre-Algebra; then you teach it again for the next two years running.
algebra without the story problems
My other theory is that Pre-Algebra is Algebra 1 without the story problems.
Algebra 1 without the story problems is, IMO, a REALLY bad idea, but that's a subject for another post.
Pre-algebra is simply arithmetic with one new feature: we use letters to represent numbers. Because the letters are simply stand-ins for numbers, arithmetic is carried out exactly as it is with numbers. In particular, the arithmetic properties (commutative, associative, distributive) hold because we are still doing arithmetic with numbers. Thus the identity
3(x + 1) = 3x + 3
holds because we know that it is true when x = 2, when x = 5, and in fact when x is any number at all.
That's it — that's all there is to prealgebra from a purely mathematical standpoint. Later, when students progress to Algebra, this basic idea is used to define functions; as algebra continues it becomes increasingly focused on functions. The purpose of prealgebra is to prepare students for variables and functions without actually mentioning them. It is a crucial topic in the middle grades.
*This is the 'grade A' review for SRA Math, the series Irvington abandoned for TRAILBLAZERS, on mathematicallycorrect. There's a dissenting review by David Klein & Jennifer Marple, (pdf file) also on mathematically correct, saying Saxon is better. Speaking of confusion, I had a memory of mathematicallycorrect reporting that Prentice-Hall Pre-Algebra had been adopted by CA; then a week ago I found a page there saying it had been rejected. Apparently I was right the first time. Was it accepted in one cycle & rejected in another? Don't know.
Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics.
So the pre-algebra course overlaps with algebra to the extent that operations are performed on symbols rather than numbers.
I think pre-algebra is just a name that's been stuck on a course regardless of whether it makes perfect logical sense or not.
The LA Times is running a story about the high rate of High School dropouts in the LAUSD.
Below are excerpts from their second article -- on the algebra requirement.
The whole series is avaailable at but may require site registration.
January 30, 2006
THE VANISHING CLASS
A Formula for Failure in L.A. Schools
By Duke Helfand, Times Staff Writer
Each morning, when Gabriela Ocampo looked up at the chalkboard in her ninth-grade algebra class, her spirits sank.
Gabriela failed that first semester of freshman algebra. She failed again and again — six times in six semesters. And because students in Los Angeles Unified schools must pass algebra to graduate, her hopes for a diploma grew dimmer with each F.
"It triggers dropouts more than any single subject," said Los Angeles schools Supt. Roy Romer. "I think it is a cumulative failure of our ability to teach math adequately in the public school system."
In the fall of 2004, 48,000 ninth-graders took beginning algebra; 44% flunked, nearly twice the failure rate as in English. Seventeen percent finished with Ds.
Among those who repeated the class in the spring, nearly three-quarters flunked again.
The school district could have seen this coming if officials had looked at the huge numbers of high school students failing basic math.
Discouragement, Frustration
Birmingham High in Van Nuys......has a failure rate that's about average for the district. Nearly half the ninth-grade class flunked beginning algebra last year.
In the spring semester alone, more freshmen failed than passed. The tally: 367 Fs and 355 passes, nearly one-third of them Ds.
Like other schools in the nation's second-largest district, Birmingham High deals with failing students by shuttling them back into algebra, often with the same teachers.
Last fall, the school scheduled 17 classes of up to 40 students each for those repeating first-semester algebra.
The strategy has also failed to provide students with what they need most: a review of basic math.
Teachers complain that they have no time for remediation, that the rapid pace mandated by the district leaves behind students like Tina Norwood, 15, who is failing beginning algebra for the third time.
Tina, who says math has mystified her since she first saw fractions in elementary school, spends class time writing in her journal, chatting with friends or snapping pictures of herself with her cellphone.
Her teacher, George Seidel, devoted a class this fall to reviewing equations with a single variable, such as x -- 1 = 36. It's the type of lesson students were supposed to have mastered in fourth grade.
"I got through a year of Vietnam," he said, "so I tell myself every day I can get through 53 minutes of fifth period…. I don't know if I am making a difference with a single kid."
Eager to close this competitive chasm, education and business leaders in California sought to re-engineer the state's approach to math. They produced new math standards they believed would foster a "rising tide of excellence."
This meant teaching algebra earlier, as soon as eighth grade for some students, even if instructors questioned whether younger students could handle abstract concepts.
'I Give Up'
Whether requiring all students to pass algebra is a good idea or not, two things are clear: Schools have not been equipped to teach it, and students have not been equipped to learn it.
Secondary schools have had to rapidly expand algebra classes despite a shortage of credentialed math teachers.
The Center for the Future of Teaching & Learning in Santa Cruz found that more than 40% of eighth-grade algebra teachers in California lack a math credential or are teaching outside their field of expertise; more than 20% of high school math teachers are similarly unprepared.
High school math instructors, meanwhile, face crowded classes of 40 or more students — some of whom do not know their multiplication tables or how to add fractions or convert percentages into decimals.
Birmingham teacher Steve Kofahl said many students don't understand that X can be an abstract variable in an equation and not just a letter of the alphabet.
Birmingham math coach Kathy De Soto said she was surprised to find something else: students who still count on their fingers.
High school teachers blame middle schools for churning out ill-prepared students. The middle schools blame the elementary schools, where teachers are expected to have a command of all subjects but sometimes are shaky in math themselves.
At Cal State Northridge, the largest supplier of new teachers to Los Angeles Unified, 35% of future elementary school instructors earned Ds or Fs in their first college-level math class last year.
Some of these students had already taken remedial classes that reviewed high school algebra and geometry.
"I give up. I'm not good at math," said sophomore Alexa Ganz, 19, who received a D in math last semester even after taking two remedial courses. "I think I've been more confused this semester than helped."
Ganz, who wants to teach third grade, thinks the required math courses are overkill. "I guarantee I won't need to know all this," she said, perhaps not realizing that if she were to teach in a public school, she could be bumped as a newcomer to upper grade levels that demand greater math knowledge.
Administrators in L.A. Unified say they are trying to reverse the alarming failure rates of high school students by changing the way math is taught, starting in elementary schools.
The new approach stresses conceptual lessons rather than rote memorization, a change that some instructors think is wrong. New math coaches also are training teachers and coordinating lesson plans at many schools.
The simplest algebraic concepts are now taught — or are supposed to be taught — beginning in kindergarten.
Searching for a solution in its secondary schools, L.A. Unified is investing millions of dollars in new computer programs that teach pre-algebra, algebra and other skills.
Officials are considering other costly changes, including reducing the size of algebra classes to 25, launching algebra readiness classes for lagging eighth-graders and creating summer programs for students needing a kick-start before middle school or high school.
Go figure
Algebra test
A majority of ninth-graders in Los Angeles fail algebra or pass with a D grade.
A copy of the LAUSD pacing plan for "Algebra I: Structure and Method" is included at the end of this letter. It undermines the organization of ideas in this textbook, and it undermines the California Mathematics Framework and Standards. We illustrate with some examples.
The two year pacing plan for "Structure and Method" calls for the quadratic formula together with completing the square of quadratic polynomials (in Chapter 12 of the textbook) to be explained to students before basic factoring techniques for polynomials (in Chapter 5), and before an introduction to radicals, including techniques for simplifying radicals (in Chapter 11). This choice of ordering of topics is so mathematically unsound that it will most likely seriously undermine the ability of LAUSD math teachers to teach algebra in a coherent and meaningful way. It will reduce the learning of algebra to memorizing meaningless formulas without understanding. In the words of the chairman of the Math Department of San Pedro High School (in LAUSD), Richard Wagoner:
"First and foremost among the problems with the Pacing Plan is that
it renders the textbook and all support materials useless and obsolete.
The sequence is so seemingly random that students must jump back and
forth between sections of chapters through most of the course.
Therefore, all review materials, diagnostic tests, supplementary
materials and enriched materials -- all of which are tied to the
sequence as designed by the authors -- can no longer be used."
Teacher George Seidel left a 25-year law career two years ago, hoping to find fulfillment as a teacher at Birmingham High in Van Nuys. "I got through a year of Vietnam," he says, "so I tell myself I can get through 53 minutes of period five.... I don't know if I am making a difference with a single kid."
Officials are considering other costly changes, including reducing the size of algebra classes to 25, launching algebra readiness classes for lagging eighth-graders and creating summer programs for students needing a kick-start before middle school or high school. Go figure
Seems like they're trying everything but what works -- teaching elementary math to mastery before teaching algebraic concepts. It is exeedingly difficult to be this wrong accidentally.
After viewing the photo gallary, I just want to say OMG. This article should be sent to every school board member pushing for "fuzzy" math.
I am a homeschooler, so I don't have a dog in this fight, but ISeeing students in these photos who have been academically-abused for the first 8 years of their school lives just floors me. Really, this country doesn't need to worry about terrorist, we are destroying our country from the inside out.
I look at those pictures and captions and everything points to how grades K-8 failed these kids. Lower schools do not want to hold kids back. One reason given for this is some study showing a higher drop-out rate for those kids who were held back. Duh! Keep passing them right along and you will have a 100% graduation rate.
At our public schools, they use the concepts of developmentally appropriate, full inclusion, and spiraling to justify this approach. They don't want classes/subjects to be filters. However, the filters are delayed until high school so they never see the product of their work. It's easy enough to blame the child, the parent, or society. By eighth grade many kids will decide that they are just not good in math, thereby completely letting the lower schools off the hook. High schools let the lower schools off the hook. They should be yelling and screaming at the lower schools.
My reaction is to tell people to look at the state tests and tell me what is so difficult that schools cannot succeed for most kids no matter what their parents or society does.
I(I'M WRITING COMMENT TO POST AS A KEY WORD SO I CAN FIND THIS AGAIN)
I had no idea homeschoolers didn't struggle over finding a math curriculum.
Of course, I didn't struggle.
There were two choices (that I knew of at the time), and I made the choice quickly once I saw both.
Here are all these teachers AND students saying 'He still counts on his fingers' — how can anyone NOT take seriously the absolute need to learn the procedures and algorithms of elementary math COLD.
My kindergarten aged son gets embarrassed when he has to resort to counting on his fingers to solve an addition or subtraction problem. You think these kids don't know that thye're not supposed to be counting on their fingers in high school? I'm sure this is all part of the low self-esteem problem you get when you you can't do math.
When you're working efficiently at your child's pace, rather than at the pace of the class & the school with its many holidays & moments of lost instructional time, I think it would be easy to get through one Saxon book & two Singapore books each school year K-5 or even K-8.
Because these kids - these young adults - really are suffering. You can see it in their body language.
This didn't need to happen aren know.
Because we are looking at pain.
These young people break my heart.
Every one of these young people can solve an equation like
3x + 2 = 32
Christopher may be able to do that now.
IF CHRISTOPHER WERE USING SAXON MATH HE'D BE ABLE TO DO THIS WITH EASE.
The only reason Christopher has found equations like this one hard is that the course moves way too fast with way too little practice.
I suppose the crafty teacher could secretly teach elementary math in class instead of algebra. It's not like these kids are in any condition to pass algebra anyway. Of course, thne this teacher looks bad since his students are all failing algebra and some other teacher will get all the glory when they ultimately pass algebra arenBut what is it?
................
My MIL taught math to middle school kids for 25 years. She.
She
"Most students who enter my eighth-grade Algebra I or Honors Algebra I classes in September each year are ill-prepared to learn algebra because most of them have not fully mastered arithmetic. To make matters worse, I have too few class periods to teach them the entire rigorous course when one adds up the drug education activities, annual class trips, report card day, vacations, snow days, exams, and parent-teacher conferences. These restrictions demand that the students put in extensive quality time outside of class grappling with difficult problems and practicing for accuracy.
They really should only count full days with the teacher when they're adding up the 180 days of school required. It would probably shock parents how few days in the year are actually spent with their teacher learning about core subjects.
"It would probably shock parents how few days in the year are actually spent with their teacher learning about core subjects."
Even with the teacher instructional time is a shrinking commodity. The first activity of the day is handing out meal ticket. That often takes half an hour of prime time. I've known teachers who schedule an hour of SSR in the morning. More prime time lost. Then you have the dogma of non-instruction under the constructivist regime. Time filled with often trivial activities and projects |
This calculator features a side window that pops out with additional tools such as trigonometry functions. It also supports stats, combinatorics and albebric functions. It includes multiple memories and simultaneous base conversion in base 2,8,10 and 16. |
Description:
by George Chalamandaris The purpose of this paper is to develop certain relatively recent mathematical discoveries known generally as stochastic calculus, or more specifically as It Discuss this paper
(No registration is required) |
This course increases maths confidence and capability and is ideal for students who are C/D borderline and need to ensure a C or boost their grade to a B.
Students join a tutor in a group to match their year group and ability, and with our interactive lessons we are able to ensure every student is engaged and progressing.
Each course syllabus will vary according to exam board although most courses will cover linear inequalities, pythagoras, trigonometry, loci, change of a subject formula, shapes and all important exam practice. For specific information on what the lessons will include, please contact us on 0845 038 0017
What
1 x 1 hour lesson per week (10 weeks in total)
When
Starts 8th, 9th, 10th and 11th April
Time
Year 10: From 6:30-7:30pm on Wed or 5:00-6:00pm on Thu
Year 11: From 6:30-7:30pm on Mon or 5:00-6:00pm on Tue
Cost
£150 (inc VAT)
Times and dates are shown in the 'available course dates' panel on the right. |
Mathematics Introduction
Here at Horndean Technology College we understand both the importance and relevance of maths in today's society. Students need to have skills in number, algebra, shape, data handling and problem solving.
All students at Key Stage 3 have 7 lessons of maths per fortnight. We follow a course based on the National Curriculum with most students sitting their Key Stage 3 SATs at the end of Year 9. However, some students are identified as being able to study a more accelerated curriculum and will sit their SATs at the end of Year 8.
At Key Stage 4 students will follow a GCSE course with most sitting their GCSE exam at the end of Year 11. Some, however, will sit their GCSE and the end of Year 10.
The department has 8 classrooms, all equipped with projectors and interactive whiteboards enabling students to benefit from all the latest software. The department also has a dedicated ICT suite allowing students to access the internet, work on the various ICT packages we have purchased and complete any project based work. |
Program that include this Course:
This course will provide the student with a working knowledge of all math formulas and equations relative to patient care in the pre-hospital emergency environment. Students will participate in mathematics and fractions review, learn systems of measurement, and drug dosage calculations in for the non-emergency and emergency environments. |
10 Units 1000 Level Course
Available in 2012
Exposes the student to a broad range of elementary but important topics in Mathematics that are especially relevant for intending early childhood and primary teachers. Topics include number concepts, elementary geometry, measurement, probability and basic statistics.
MATH1900 cannot be counted for credit with MATH1110 or MATH1210 unless taken before these courses. MATH1900 cannot count for credit with MATH1100 nor MATH1410. |
Level:(8) Eighth Grade Course Title: Algebra Readiness Course Description: (Algebra Readiness A/B is a course to better prepare students to approach the state and college required Algebra 1 A/B course. Many pupils in Eagle Rock Junior High come into eighth grade lacking the mathematics skills to understand and pass the algebra course. This course has been offered for students who have had difficulty in math in the past to increase their mathematics skills. It is an intermediate step for many students to avoid failure in Algebra.
MYP Aims Addressed by this Course: The aims of teaching and learning mathematics are to encourage and enable students to:
recognize that mathematics permeates the world around us
appreciate the usefulness, power and beauty of mathematics
enjoy mathematics and develop patience and persistence when solving problems
understand and be able to use the language, symbols and notation of mathematics
develop mathematical curiosity and use inductive and deductive reasoning when solving problems
become confident in using mathematics to analyze and solve problems both is school and in real-life situations
develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics
develop abstract, logical and critical thinking and the ability to reflect critically upon their work and the work of others
develop a critical appreciation of the use of the information and communication technology (ICT) in mathematics
appreciate the international dimension of mathematics and its multicultural and historical perspectives
Time (weeks)
Instructional Units for the Year
Essential and/or Guiding Questions
Area of Interaction
Assessment
2 - 3
2 - 3
4 - 8
3 - 4
4 - 6
8- 10
Variables, Expressions and properties
Integers and Equations
Fractions and decimals
Exponents and Roots
Ratios, Rates, Proportion and Percent
Algebra on the Coordinate Plane
What are variables? What are expressions? How may I solve for a variable? How many different ways can I write an expression?
How do I handle the data from my environment? How do I group the subsets in my environment?
Why were fractions necessary to invent? What are the consequences of using fractions and decimals? Why were decimals developed after fractions?
How can a disease spread? How can we handle large numbers so they are understandable? How can we safeguard ourselves and others?
What resources do we need and have? How do we measure our usage?
Where do we live? How do we live in relation to each other? How can we locate ourselves in our community?
Approaches to learning
Environment
Homo faber
Health and Social Education
Environment
Community and Service
The publisher provides quizzes and tests for each section of their product. Teacher made tests will also be given. Projects assigned Criterion A: Knowledge and Understanding
The publisher provides quizzes assigned. Criterion A: Knowledge and Understanding
The publisher provides quizes and group work assigned. Criterion A: Knowledge and Understanding Criterion D: Reflection in mathematics
The publisher provides quizzes and tests for each section of their product. Teacher made tests will also be given. Criterion A: Knowledge and Understanding Criterion D: Reflection in mathematics |
Solving
equations
Once we can solve quadratic equations and simultaneous equations it is tempting to think that we will be able to solve more and more complicated equations by using the same principles but its not always that simple. The
quadratic equation:
a
x² + b x + c = 0
can
be solved using a formula but most polynomial equations don't have
formulas to give the solution. Every polynomial with only real or complex coefficients has a complex number solution , this is the fundamental theorem of algebra, but they cannot always be expressed exactly with radicals. To find an approximation to the solution we may have to use numerical methods such as the Newton-Raphson method or Laguerre method.
Generalised reciprocity laws are very
complicated algorithms that enable you to get crucial information
about some of these more complicated equations. In favorable
circumstances, they can be used to prove deep statements about the
solution sets of algebraic equations.
An
equation does not have to be too complicated to lack a formula, for
example:
x5 - x + 1 = 0
Numbers
- Real
numbers - numbers which are continuous such as when we are representing points along a line - On this site I will sometimes use the term 'Scalar' to mean 'Real' numbers although strictly the term should be used when scaling a vector - In computer programs real numbers have a finite length and may have decimal point and/or exponent this allows us to approximate most real numbers but it is only an approximation.
Set Definition
Function Definition
A
function from a set A to a set B is a rule that assigns to each
element in A an element of B. If f is the name of the function and a is an element of A then we write f(a) to mean the
element of B that is assigned to a. A function f is often written as f: A →B.
Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them.
Fearless Symmetry - This
book approaches symmetry from the point of view of number theory. It
may not be for you if you are only interested in the geometrical
aspects of symmetry such as rotation groups but if you are interested
in subjects like modulo n numbers, Galois theory, Fermats last
theorem, to name a few topics the chances are you will find this book
interesting. It is written in a friendly style for a general audience
but I did not find it dumbed down. I found a lot of new concepts to
learn. It certainly gives a flavor of the complexity of the subject
and some areas where maths is still being discovered |
An online course offering Java applets and 3D graphics to teach 3D vectors and related topics such as 3D graphics programming and Newtonian mechanics. Contents: 3D Vectors; The Dot Product; The Cross Product; the Vector Equation of a Line; The Vector Equation of a Plane; The Point of Intersection of a Line and a Plane; Some Shortest Distance Calculations; Coordinate Systems; and An introduction to parametric curves. Test what you have learned by finding the distance between two cities along the surface of the Earth, docking at a space station, or calculating the angle between two adjacent sides of a dodecahedron. |
MathSkills reinforces math in three key areas: pre-algebra, geometry, and algebra. These titles supplement any math textbook. Reproducible pages can be used in the classroom as lesson previews or reviews. The activities are also perfect for homework or end-of-unit quizzes. |
This program illustrates functions for solving systems of linear and quadratic equations. Using matrices, students solve equations in a time-efficient manner. A chef shows how mathematics keeps things cooking at his restaurant. |
Mathematical Proofs A Transition to Advanced Mathematics
9780321390530
ISBN:
0321390539
Edition: 2 Pub Date: 2007 Publisher: Pearson Addison-Wesley
Summary: Mathematical Proofs: A Transition to Advanced Mathematics, 2/e,prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.KEY TOPICS: Communicating Mathematics, Sets, Logi...c, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory.MARKET: For all readers interested in advanced mathematics and logic |
Incorporating recent developments that have made it possible to solve difficult optimization problems with greater accuracy, author Laurence A. Wolsey presents a number of state-of-the-art topics not covered in any other textbook. These include improved modeling, cutting plane theory and algorithms, heuristic methods, and branch-and-cut and integer programming decomposition algorithms. This self-contained text:
* Distinguishes between good and bad formulations in integer programming problems
* Applies lessons learned from easy integer programs to more difficult problems
* Demonstrates with applications theoretical and practical aspects of problem solving
* Includes useful notes and end-of-chapter exercises
* Offers tremendous flexibility for tailoring material to different needs |
College Algebra With Trigonometry
9780073312644
ISBN:
0073312649
Edition: 8 Pub Date: 2007 Publisher: McGraw-Hill College
Summary: The Barnett, Ziegler, Byleen College Algebra series is designed to be user friendly and to maximize student comprehension. The goal of this series is to emphasize computational skills, ideas, and problem solving rather than mathematical theory. College Algebra with Trigonometry, 7/E, introduces a right angle approach to trigonometry and can be used in one or two semester college algebra with trig or precalculus cours...es. The large number of pedagogical devices employed in this text will guide a student through the course. Integrated throughout the text, the students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematical concepts. In each section, the worked examples are followed by matched problems that reinforce the concept that is being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A Smart CD is packaged with the seventh edition of the book. This CD reinforces important concepts, and provides students with extra practice |
Some basic concepts of Mathematics
(that we will frequently use.. read this section of
your book carefully)
Functions
Variables ..Discrete and Continuous (based on nature of a variable
taking values)..examples in Economics?
-dependent variable and independent variable (based on
relationship)..examples? Little bit problematic sometimes?
Positive relationship – negative relationship
straight line Vs Curved lines
Slope as a measurement of effect of a change in one variable..
some kind of 'partial' sense (more on higher level courses)
Continued..
What happens when any other independent variables
change? …idea of 'Shift' ?
Shifts Vs Movements along the line
Solving equations
No more on Computer – let's use the blackboard. |
Fourier Series Concept Visualization Tools
Chris Long
For many students the concepts associated with Fourier series can be difficult to comprehend. Two of those concepts are the principle of superposition and the relationship between the time and frequency domains. At Purdue South Bend, Professor Gene Harding and undergraduate student Chris Long worked together using MATLAB to create a pair of visualization tools for illustrating these concepts. Chris did a great job developing these tools, which were used for the first time in ECET207 during the fall 2007 semester. They were subsequently adopted by the main campus and a local community college.
The links for the animations and associated m-files are below. To view them just click on the links. If you want to save a file, right-click on the link, select Save Target As..., and enter a location and filename where you want it to be stored on your computer.
Feedback or questions about the project are welcome. Please direct them to Professor Harding. |
Mathematical Communication
Wording and punctuation
This webpage lists resources for helping students to learn the wording and punctuation conventions of mathematics. Resources include online and print math dictionaries, handouts that include notes on word choice, etc. |
Because of its large command structure and intricate syntax, Mathematica can be difficult to learn. Wolfram's Mathematica manual, while certainly comprehensive, is so large and complex that when trying to learn the software from scratch - or find answers to specific questions - one can be quickly overwhelmed.
A Beginner's Guide to Mathematica offers a simple, step-by-step approach to help math-savvy newcomers build the skills needed to use the software in practice. Concise and easy to use, this book teaches by example and points out potential pitfalls along the way. The presentation starts with simple problems and discusses multiple solution paths, ranging from basic to elegant, to gradually introduce the Mathematica toolkit. More challenging and eventually cutting-edge problems follow. The authors place high value on notebook and file system organization, cross-platform capabilities, and data reading and writing. The text features an array of error messages you will likely encounter and clearly describes how to deal with those situations.
While it is by no means exhaustive, this book offers a non-threatening introduction to Mathematica that will teach you the aspects needed for many practical applications, get you started on performing specific, relatively simple tasks, and enable you to build on this experience and move on to more real-world problems.
"This book is a comprehensive package for knowledge sharing on Mathematics. The language of the book is simple and self-explanatory, this will help the students to grasp the fundamentals of the subject easily. The book follows a to the point approach and lays stress on the understanding of the core concepts. Appropriate number of MCQs are given for each topic that are of great help to the students appearing for competitive and State Board examinations.""
" . . . [a] treasure house of material for students and teachers alike . . . can be dipped into regularly for inspiration and ideas. It deserves to become a classic." --London Times Higher Education Supplement "The author succeeds in his goal of serving the needs of the undergraduate population who want to see mathematics in action, and the mathematics used is extensive and provoking." --SIAM Review "Each chapter discusses a wealth of examples ranging from old standards . . . to novelty . . . each model is developed critically, analyzed critically, and assessed critically." --Mathematical Reviews A Concrete Approach to Mathematical Modelling provides in-depth and systematic coverage of the art and science of mathematical modelling. Dr. Mesterton-Gibbons shows how the modelling process works and includes fascinating examples from virtually every realm of human, machine, natural, and cosmic activity. Various models are found throughout the book, including how to determine how fast cars drive through a tunnel, how many workers industry should employ, the length of a supermarket checkout line, and more. With detailed explanations, exercises, and examples demonstrating real-life applications in diverse fields, this book is the ultimate guide for students and professionals in the social sciences, life sciences, engineering, statistics, economics, politics, business and management sciences, and every other discipline in which mathematical modelling plays a role.
Most texts on nonparametric techniques concentrate on location and linear-linear (correlation) tests, with less emphasis on dispersion effects and linear-quadratic tests. Tests for higher moment effects are virtually ignored. Using a fresh approach, A Contingency Table Approach to Nonparametric Testing unifies and extends the popular, standard tests by linking them to tests based on models for data that can be presented in contingency tables.
This approach unifies popular nonparametric statistical inference and makes the traditional, most commonly performed nonparametric analyses much more complete and informative. It also makes tied data easily handled, and almost exact Monte Carlo p-values can be obtained. With data in contingency tables, one can then calculate a Pearson-type, chi-squared statistic and its components. For univariate data, the initial tests based on these components detect mean differences between treatments. For bivariate data, they detect correlations. This approach leads to tests that detect variance, skewness, and higher moment differences between treatments with univariate data, and higher bivariate moment differences with bivariate data.
Although the methods advanced in this book have their genesis in traditional nonparametrics, incorporating the power of modern computers makes the approach more complete and more valid than previously possible. The authors' unified treatment and readable style make the subject easy to follow and the techniques easily implemented, whether you are a fledgling or a seasoned researcher.
The first contemporary textbook on ordinary differential equations (ODEs) to include instructions on MATLAB®, Mathematica®, and Maple™, A Course in Ordinary Differential Equations focuses on applications and methods of analytical and numerical solutions, emphasizing approaches used in the typical engineering, physics, or mathematics student's field of study. Stressing applications wherever possible, the authors have written this text with the applied math, engineer, or science major in mind. It includes a number of modern topics that are not commonly found in a traditional sophomore-level text. For example, Chapter 2 covers direction fields, phase line techniques, and the Runge-Kutta method; another chapter discusses linear algebraic topics, such as transformations and eigenvalues. Chapter 6 considers linear and nonlinear systems of equations from a dynamical systems viewpoint and uses the linear algebra insights from the previous chapter; it also includes modern applications like epidemiological models. With sufficient problems at the end of each chapter, even the pure math major will be fully challenged. Although traditional in its coverage of basic topics of ODEs, A Course in Ordinary Differential Equations is one of the first texts to provide relevant computer code and instruction in MATLAB, Mathematica, and Maple that will prepare students for further study in their fields. for time series model building. The authors begin with basic concepts in univariate time series, providing an up-to-date presentation of ARIMA models, including the Kalman filter, outlier analysis, automatic methods for building ARIMA models, and signal extraction. They then move on to advanced topics, focusing on heteroscedastic models, nonlinear time series models, Bayesian time series analysis, nonparametric time series analysis, and neural networks. Multivariate time series coverage includes presentations on vector ARMA models, cointegration, and multivariate linear systems. Special features include: Contributions from eleven of the world??'s leading figures in time series Shared balance between theory and application Exercise series sets Many real data examples Consistent style and clear, common notation in all contributions 60 helpful graphs and tables Requiring no previous knowledge of the subject, A Course in Time Series Analysis is an important reference and a highly useful resource for researchers and practitioners in statistics, economics, business, engineering, and environmental analysis. An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department.
Developed from the authors, combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems governed by partial differential equations. Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, the book is both introductory and self-contained, as well as being a hands-on experience for any student. This authoritative text on Finite Elements: Adopts a generic approach to the subject, and is not application specific In conjunction with a web-based chapter, it integrates code development, theory, and application in one book Provides an accompanying Web site that includes ABAQUS Student Edition, Matlab data and programs, and instructor resources Contains a comprehensive set of homework problems at the end of each chapter Produces a practical, meaningful course for both lecturers, planning a finite element module, and for students using the text in private study. Accompanied by a book companion website housing supplementary material that can be found at A First Course in Finite Elements is the ideal practical introductory course for junior and senior undergraduate students from a variety of science and engineering disciplines. The accompanying advanced topics at the end of each chapter also make it suitable for courses at graduate level, as well as for practitioners who need to attain or refresh their knowledge of finite elements through private study.
Although the use of fuzzy control methods has grown nearly to the level of classical control, the true understanding of fuzzy control lags seriously behind. Moreover, most engineers are well versed in either traditional control or in fuzzy control-rarely both. Each has applications for which it is better suited, but without a good understanding of both, engineers cannot make a sound determination of which technique to use for a given situation.
A First Course in Fuzzy and Neural Control is designed to build the foundation needed to make those decisions. It begins with an introduction to standard control theory, then makes a smooth transition to complex problems that require innovative fuzzy, neural, and fuzzy-neural techniques. For each method, the authors clearly answer the questions: What is this new control method? Why is it needed? How is it implemented? Real-world examples, exercises, and ideas for student projects reinforce the concepts presented.
Developed from lecture notes for a highly successful course titled The Fundamentals of Soft Computing, the text is written in the same reader-friendly style as the authors' popular A First Course in Fuzzy Logic text. A First Course in Fuzzy and Neural Control requires only a basic background in mathematics and engineering and does not overwhelm students with unnecessary material but serves to motivate them toward more advanced studies.
The field of applied probability has changed profoundly in the past twenty years. The development of computational methods has greatly contributed to a better understanding of the theory. A First Course in Stochastic Models provides a self-contained introduction to the theory and applications of stochastic models. Emphasis is placed on establishing the theoretical foundations of the subject, thereby providing a framework in which the applications can be understood. Without this solid basis in theory no applications can be solved.Provides an introduction to the use of stochastic models through an integrated presentation of theory, algorithms and applications.Incorporates recent developments in computational probability.Includes a wide range of examples that illustrate the models and make the methods of solution clear.Features an abundance of motivating exercises that help the student learn how to apply the theory.Accessible to anyone with a basic knowledge of probability.A First Course in Stochastic Models is suitable for senior undergraduate and graduate students from computer science, engineering, statistics, operations resear ch, and any other discipline where stochastic modelling takes place. It stands out amongst other textbooks on the subject because of its integrated presentation of theory, algorithms and applications.
Realizing the specific needs of first-year graduate students, this reference allows readers to grasp and master fundamental concepts in abstract algebra-establishing a clear understanding of basic linear algebra and number, group, and commutative ring theory and progressing to sophisticated discussions on Galois and Sylow theory, the structure of abelian groups, the Jordan canonical form, and linear transformations and their matrix representations.
R is dynamic, to say the least. More precisely, it is organic, with new functionality and add-on packages appearing constantly. And because of its open-source nature and free availability, R is quickly becoming the software of choice for statistical analysis in a variety of fields.
Doing for R what Everitt's other Handbooks have done for S-PLUS, STATA, SPSS, and SAS, A Handbook of Statistical Analyses Using R presents straightforward, self-contained descriptions of how to perform a variety of statistical analyses in the R environment. From simple inference to recursive partitioning and cluster analysis, eminent experts Everitt and Hothorn lead you methodically through the steps, commands, and interpretation of the results, addressing theory and statistical background only when useful or necessary. They begin with an introduction to R, discussing the syntax, general operators, and basic data manipulation while summarizing the most important features. Numerous figures highlight R's strong graphical capabilities and exercises at the end of each chapter reinforce the techniques and concepts presented. All data sets and code used in the book are available as a downloadable package from CRAN, the R online archive.
A Handbook of Statistical Analyses Using R is the perfect guide for newcomers as well as seasoned users of R who want concrete, step-by-step guidance on how to use the software easily and effectively for nearly any statistical analysis.
WILEY-INTERSCIENCE PAPERBACK SERIESFrom the Reviews of History of Probability and Statistics and Their Applications before 1750"This is a marvelous book . . . Anyone with the slightest interest in the history of statistics, or in understanding how modern ideas have developed, will find this an invaluable resource."--Short Book Reviews of ISI
Because of its portability and platform-independence, Java is the ideal computer programming language to use when working on graph algorithms and other mathematical programming problems. Collecting some of the most popular graph algorithms and optimization procedures, A Java Library of Graph Algorithms and Optimization provides the source code for a library of Java programs that can be used to solve problems in graph theory and combinatorial optimization. Self-contained and largely independent, each topic starts with a problem description and an outline of the solution procedure, followed by its parameter list specification, source code, and a test example that illustrates the usage of the code.
The book begins with a chapter on random graph generation that examines bipartite, regular, connected, Hamilton, and isomorphic graphs as well as spanning, labeled, and unlabeled rooted trees. It then discusses connectivity procedures, followed by a paths and cycles chapter that contains the Chinese postman and traveling salesman problems, Euler and Hamilton cycles, and shortest paths. The author proceeds to describe two test procedures involving planarity and graph isomorphism. Subsequent chapters deal with graph coloring, graph matching, network flow, and packing and covering, including the assignment, bottleneck assignment, quadratic assignment, multiple knapsack, set covering, and set partitioning problems. The final chapters explore linear, integer, and quadratic programming. The appendices provide references that offer further details of the algorithms and include the definitions of many graph theory terms used in the book. |
I'm starting a new job soon where I will be using Matlab a whole lot. I've used Matlab (and Octave) before and know the basics, but I want to become an expert. What book should I buy, or what other resources should I look into? [more inside]
posted by no regrets, coyote
on Jan 27, 2012 -
7 answers
I have always been horrible at math, but somehow a great programmer. I have found that writing a computer program that demonstrates a certain mathematical concept enables me to better understand the concept. I'm a psych major and I brought this up once in the research lab I've been working in. My prof said he recalls that someone did research and/or created a system in which a student writes a computer program that is pertinent to a certain mathematical concept and upon completion is given the regular math problem (as it would appear in a math class). This enables the student to better understand the math problem, solve, and learn math. Has anyone heard of this or anything similar? A learning system such as this would be a blessing to my education.
Thanks.
posted by fightoplankton
on Apr 13, 2009 -
15 answers |
Discovering Advanced Algebra
Overview
Develop your students' understanding of abstract math concepts with Discovering Advanced Algebra, a research-based, CCSS-aligned Algebra 2 program that features real-world applications that resonate with all learners. Click here for a Discovering Mathematics series overview. To sign up for a free 30-day online trial, click here. |
Problem Solving with the TI-83/TI-83 Plus/84 Plus
This book is a rich collection of original problems designed to make use of the advanced features of the TI-83/84 to explore topics in algebra, geometry, precalculus, calculus, finance, statistics and probability.
The solutions exhibit the power of calculators in general and the TI-83/84 specifically. The reader will learn the operation of the TI-83/84 by solving interesting problems and not just pressing keys. It is for students of mathematics of any age, and is an excellent resource for the classroom or for teacher training. |
books.google.ca - Teaching Secondary Mathematics, Third Edition is practical, student-friendly, and solidly grounded in up-to-date research and theory. This popular text for secondary mathematics methods courses provides useful models of how concepts typically found in a secondary mathematics curriculum can be delivered... Secondary Mathematics
Teaching Secondary Mathematics
Teaching Secondary Mathematics, Third Edition is practical, student-friendly
A variety of approaches, activities, and lessons is used to stimulate the reader's thinking--technology, reflective thought questions, mathematical challenges, student-life based applications, and group discussions. Technology is emphasized as a teaching tool throughout the text, and many examples for use in secondary classrooms are included. Icons in the margins throughout the book are connected to strands that readers will find useful as they build their professional knowledge and skills: Problem Solving, Technology, History, the National Council of Teachers of Mathematics Principles for School Mathematics, and "Do" activities asking readers to do a problem or activity before reading further in the text. By solving problems, and discussing and reflecting on the problem settings, readers extend and enhance their teaching professionalism, they become more self-motivated, and they are encouraged to become lifelong learners.
New in the Third Edition: *All chapters have been thoroughly revised and updated to incorporate current research and thinking. *The National Council of Teachers of Mathematics Standards 2000 are integrated throughout the text. *Chapter 5, Technology, has been rewritten to reflect new technological advances. *A Learning Activity ready for use in a secondary classroom has been added to the end of each chapter. *Two Problem-Solving Challenges with solutions have been added at the end of each chapter. *Historical references for all mathematicians mentioned in the book have been added within the text and in the margins for easy reference. *Updated Internet references and resources have been incorporated to enhance the use of the text.
About the author (2006)
Douglas K. Brumbaugh has taught college, in-service, or K-12 mathematics for close to 45 years. He received his B.S. from Adrian College and master's and doctorate in mathematics education from the University of Georgia. When talking with others about teaching and learning in the K-12 environment, his immersion in teaching is beneficial. The thoughts and examples in this book are based experiences working with garden-variety kids. Peggy L. Moch decided to become a teacher, following nearly 25 years in the laboratory as a medical technologist mainly specializing in immunohematology, microbiology, quality control, and quality assurance. She obtained her teaching degrees, a bachelor's, master's, and doctorate--all in mathematics education--from the University of Central Florida. Together with her writing colleagues, Moch has endeavored to share that love with readers. MaryE Wilkinson received her B.S., master's, and doctorate in mathematics education from the University of Central Florida. She has taught high school mathematics courses, college mathematics courses, and mathematics and science methods courses. When thinking and writing about elementary school mathematics, she draws upon all of her experiences and continuing presence in elementary classrooms.
David Rock is Professor of History at the University of California, Santa Barbara. His "Politics of Argentina, 1890-1930 won the 1976 Herbert E. Bolton Prize for Latin American history. He is author of " Authoritarian Argentina" (1993) and editor of "Latin America in the Nineteen Forties" (1994), both available from the University of California Press. |
Math Activities Center (MAC)
The Math Activities Center (MAC) is a math study area where registered students can get "drop-in" help with math assignments. Students will find a combination of peer tutors, staff and faculty on duty to assist them all hours the MAC is open. Instructor schedules are posted in the MAC each semester. Extended one-on-one tutoring is available free of charge by appointment at the Tutoring Center (room 402).
Textbooks and Solution Manuals
The MAC maintains a collection of current textbooks and student solution manuals for most SCC math courses. They can be used during a student's time in the MAC in exchange for a Solano Student ID. Students without an ID may purchase one by following the directions here. The MAC also maintains a large collection of reference textbooks, solution manuals, videotapes, CD-ROMs, graphing and scientific calculators for use in the lab.
Computer Access
The MAC offers SCC math students access to eighteen PC workstations during all hours the Math Lab is open. These computers are loaded with a variety of math software programs, including Minitab and Maple. They also provide "MyMathLab" and ALEKS access. Each computer has a TI-83+ graphing calculator loaded on the desktop. Students using these computers must bring a USB flash drive if they wish to save their work. A color printer is available for color plots, in addition to a high speed black and white printer to print math assignments. Please note that the MAC and its computers are reserved for math use only. Internet browsing and general applications are available at the library (building 100).
Lab Requirement
Students enrolled in math classes that have a lab requirement need to spend one hour a week in the MAC. Accumulated lab time is tracked and recorded by computer as students check in and out of the lab. It is important that students remember to log in and out whenever they visit the lab. Instructors receive weekly reports of each student's accumulated hours in the MAC, and determine how those hours will be incorporated into the final grade. (Note: Hours are accumulated separately for each math class.)
Employment (Student Tutors)
The MAC employs 10 - 12 student tutors each semester. If you are an SCC student who has completed a calculus sequence with a B or better, have good communication skills and work well with people, please consider applying for a tutoring position. Applicants must obtain a recommendation from a SCC math instructor. Interested applicants should obtain an application at the front desk of the MAC. |
Find a Hazard, CA MathPractice makes perfect, it's better to use examples that will be easier for students to comprehend. Furthermore, learning is a process. Students build up concepts gradually and examples may be used several times before they can fully understandAlgebra course on the Coast Guard base in San Pedro. |
Peer review is a process of self-regulation by a profession or a process of evaluation involving qualified individuals within the relevant field. Peer review methods are employed to maintain standards, improve performance and provide credibility...
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background.
Paid circulation in 2008 was 9,500 and total circulation was 10,000. |
Designed for undergraduate students, A Mathematics Companion for Science and Engineering Students provides a valuable reference for a wise variety of topics in pre-calculus mathematics. The presentation is brief and to-the-point, but also precise, accurate and complete.
Learn how to read mathematical discourse, write mathematics appropriately, and think in a way that is conducive to solving mathematical problems. Topics covered include: Logic, sets numbers, sequence, functions, powers and roots, exponentials and logarithms, possibility, matrices, Euclidean geometry, analytic geometry, and the application of mathematics to experimental data. The epilogue introduces advanced topics from calculus and beyond. A large appendix offers 360 problems with fully detailed solutions so students can assess their basic mathematical knowledge and practice their skills.
Here are just some of the questions answered in this book: How can
a) Logarithm be converted from one base to another?
b) Simultaneous linear equations be solved by hand painlessly?
c) Some infinities be bigger then other? |
Published books of problems generally mean to nudge serious, dedicated and perhaps talented mathematics students towards a research-oriented frame of mind. If you really mean a bound problem collection for general education, I expect you will have to write such a book yourself. But I would probably recommend against publishing such a book - on the grounds: don't teach until you see the whites of their eyes. A mathematics problem that will work with one cohort might variously and unpredictably either defeat or insult the intelligence of another. And, as the the response to your other question might indicate, teachers of mathematics will have very diverse views of what constitute a value partial knowledge of mathematics.
That said, if I had an audience of highly intelligent but not especially mathematically oriented students, I might focus their "last look" at mathematics on Lawvere and Schanuel's Conceptual Mathematics (which has many good problems). The authors show themselves as both wise and smart. While the book could save the soul of a stray mathematician, it does not harbor any hidden agenda that means ignoring the needs of the broader audience. And while it might accidentally remediate some high school induces confusions, even the best trained students will find most of what the authors say both very new and very fundamental.
Serge Lang's Math Talks for Undergraduate also attracts me, but where Lawvere and Schanuel help a student think about the larger world in a more mathematical way, Lang wants non-mathematicians to understand more about what mathematicians do.
Research mathematician/teachers generally hold as a sacred shibboleth the dictum that "mathematics in not a spectator sport." In the case of a class of general education students seeing mathematics in the classroom for the last time, and and the risk of blasphemy, I question this, I question whether having these students primarily trying to solve problems for themselves necessarily constitutes the best use of their time. I believe that the mathematics community has neglected developing of literature of what one might call proof-oriented spectator mathematics. But still I might recommend a book: I taught a course recently out of Ross Honsberger's Episodes in 19th and 20th Century Euclidean Geometry where I focused on close readings of complicated but elementary proofs of concrete and yet often spectacularly counter-intuitive facts, all material most mathematics majors will never see on the grounds that it isn't sufficiently modern. But as a toy model of what mathematicians do it worked very well for my students. |
Aims and Objectives This course is an introduction to pure mathematics. If you
follow the course, you will have a grasp of contemporary mathematical notation, and
will become familiar with various methods of proof.
Need Help? You have the following options
(in order of preference): 1 think, 2 read the lecture notes, 3 read the book, 4 talk to
fellow students, 5 ask the person in charge of your tutorial group,
6 go to a workshop drop-in session, 7
go to a
MASH
drop-in session, 8 Ask nRich (see below), 9 ask your personal tutor (unless you have one of
the odd ones who wants to discuss your life)
and if all else fails, 10 make an
appointment with me by e-mail.
The best bulletin board that supports undergraduate mathematicians is
nRich. You register at that site, and then click on
the Ask nRich link. There are bulletin boards at various levels. The one appropriate
to undergraduates is Higher
Dimension.
You may find some early undergraduate topics being discussed in Onwards and Upwards. The latter is
really for people aged 16-18, but smart school students often discuss university mathematics.
Discussions worth reading Over at Cambridge, Tim Gowers is introducing a new feature this year,
his Weblog.
While it will be tailored to his first year lectures at his university, I expect we will all learn a thing
or two from Tim. There are Terry Tao's
maths quizzes, which seem to
be very attractive aids to learning pure maths in general (not just
for this course). Both Terry
and Tim
have Fields Medals (these are like Nobel
prizes, but for clever people).
Timetable Here is the
course timetable.
For this course, Lectures (and Problem Classes) are at
Tuesday 9:15 University Hall,
Thursday 16:15 University Hall
and Friday 12:15 (Problem Class) University Hall.
The Mathematics Department has set up drop-in Workshops to advise students
on this course (and others). These are
14:15 -- 16:05 Monday 1W4.40,
10:15 -- 12:05 Tuesday 1W4.40
and 17:15 -- 19:05 1W4.40.
You can drop in to any of them for as long, or as short, as you wish.
The workshops will not run in Week 1 but
our Algebra 1A tutorials will run in week 1.
There are also drop-in help sessions organized by MASH.
These are on Tuesday 14:15 -- 16:05 in 1W2.5,
Wednesday 13:15 -- 15:05 in 1W3.15
and Thursday 13:15 -- 15:05 in 4W1.7. Please let me know if any of this information is broken.
Feedback The best way to get feedback on your progress is via your tutor. Hand-in work by the specified
time and date in the relevant 4W level 1 slot. Your tutor will comment on your work, which should be returned
to you about a week later. It is a crime to go to a tutorial without having tried the problems
on the current sheet. It is also a crime to go to a tutorial without knowing the meaning of every word
used on the problem sheet (since if you do not understand the words, you will not understand the questions).
It is most definitely not a crime to be stuck, confused or bamboozled, and your tutor should be able to help you there.
Do not ask your tutor to show you how to do the problems. Rather your tutor should help you understand the problems
with sufficient insight that you can do some of the problems yourself. Problems come in varying levels of difficulty,
and it would be an extraordinary student who could regularly do them all.
What do you do if you think I have made an error in lectures, on a question sheet or on a solution sheet? Well, first check if someone else agrees with you. Then if so, send me an e-mail with the relevant information.
Lecturer's Lecture Notes These are in my head, and therefore cannot be borrowed. If you miss a lecture or lectures, then borrow a set of notes from a reliable scribe, and copy them up by hand or photocopying machine.
Problem Sheet Solutions Written solutions will be put up at this site
after the work has been handed in.
Lecture 1Boole's rules
are here. Problem Sheet 1. In this lecture we made three definitions and introduced
some notation. We defined the notion of a set, and wrote
x ∈ A to mean that x is an element of the set A.
We defined the notion of a subset, and wrote A ⊆ B.
We decided to write A = B when both A ⊆ B and B ⊆ A.
It follows that there is a unique set with no elements,
so we can apply the definite article to the empty set ø.
The set of natural numbers is ℕ,
the set of integers is ℤ,
the set of rational numbers is ℚ,
the set of real numbers is ℝ and the set of complex numbers is ℂ.
We introduced interval notation (a,b), [a,b], (a, b] and [a,b].
Thus, for example, (1,2) = { x | x ∈ ℝ, 1 < x, x < 2}.
Finally I asked if there was a collection of open intervals with the property
that the intersection of each pair is not the empty set, but the intersection
of the whole collection is the empty set. Incidentally, I spent
last Sunday marking the 2012 UK
Mathematical Olympiad for Girls.
Lecture 2 We introduced more notation, including that
for intersection and union. We discussed Boole's rules
at some length, including De Morgan's laws. We began to discuss maps.
A dark warning was issued against allowing
the set
of all sets to be a set, for Bertrand Russell is waiting for the
unwary, and will hit you with a
paradox if
you do that. Note the quality of the
moustache.
Lecture 3 We introduced the identity map on a set, and constant
maps. We defined composition of maps, and showed that
composition of maps, where defined, is an associative process.
We defined injective, surjective and bijective maps.
We proved that each of these last three types of map
is closed under map composition (Proposition 10).
Here is Problem Sheet 2,
due in on Monday October 17.
Here are
Solutions to sheet 1. Note the use of the "maps to" symbol, a right arrow
with a short vertical tail. This is used to describe how a map acts.
Thus f: x |-> x^2 means the same as f(x) = x^2. Sorry about the home made symbols, I am still looking for the way to display the "maps to" symbol in html.
This is not the same symbol as the right arrow which sits between the
domain and codomain of a map.
Lecture 4
We characterized injections, surjections and bijections between
non-empty sets in terms of the existence of left, respectively right, respectively (unique) two sided inverses. We introduced the notion of the power
set of a set. We showed that if A is a finite set, then |P(A)| = 2n.
We began the proof that for any set B, there is no bijection between B and
P(B).
Lecture 5.
We finished the proof mentioned above. We proved that there is a bijection between
the natural numbers and the integers. We proved that if S is an infinite subset of
the natural numbers, then there is a bijection between S and the natural numbers.
We proved that there is a bijection between the set of ordered pairs of natural numbers
and the natural numbers. We stated (but will not prove) that if A and B are sets, then either
there is an injection from A to B, or there is an injection from B to A (or possibly both).
We also stated, but will not prove in lectures, the Schroeder-Bernstein Theorem, that
if A and B are sets, and there are injections both from A to B, and from B to A, then
there is a bijection between A and B.
Problem Sheet 3 and
Sheet 2 Solutions.
Lecture 6 We defined countability: a set S is countable
if (and only if) there is an injection f : S → ℕ
We proved that the set ℚ of rational numbers is (infinite) countable.
We used Cantor's diagonal argument to show that
the real interval [0,1/2] is not countable, and so
ℝ is not countable. Note that there was a typo (chalko?) in
the diagonal argument. At one point I wrote yi
when I should have written yii.
Lecture 7 Here are
notes v2 on partitions and equivalence relations.
here are the changes to version 1 in case you have it. First line, R is a subset of S X S, not R is a subset of S.
In "Properties of Relations", the erroneous spelling reflextive was eliminated. In Examples of Partitions (iii), change "for" to "form".
In Examples of Equivalence Relations (ii), insert the missing comma after the word "sets". In the discussion after teh Examples of Transversals,
correct the mangled spelling of "equivalence". In teh final part on scary notation, change [3] to [2].
At the Problems Class on Friday October 19 2012, a student called Miles
proposed a better solution to Sheet 2, problem 8(d), than the one
which I had suggested in the solutions sheet.
Lecture 8 We started number theory. We defined prime numbers, and proved that there are infinitely many of
them. We defined coprimality, and we showed that if m and n are integers and not both 0, then
the smallest positive integer g expressible as rm + sn (with integers r and s) is the greatest common divisor of
m and n, and moreover that every common divisor of m and n will divide g.
We stated the Fundamental Theorem of Arithmetic, that every positive integer greater than 1
is the product of prime numbers in a (more or less) unique way. We got as far as proving the existence of
such a factorization, and will address uniqueness in the lecture on Thursday.
This note addresses
the chalko in Euclid's proof, and the matter of good housekeeping (induction) as promised in the
lecture.
Here are Sheet 3 Solutions.
Lecture 9 We completed the proof of Gauss's Fundamental Theorem of Arithmetic.
We discussed how to
count the number of divisors of a natural number by
looking at its factorization into prime numbers. We compared the prime factorization of positive
integers m and n with their prime factorizations.
We discussed Euclid's algorithm, why it terminates, and why it gives the gcd of two positive integers as the output.
Here is a question which has a nice answer: "what is the sum of the divisors of 1000?". You can do it in your head
(provided that you are relaxed about multiplying a three-digit number by a two-digit number, and who isn't?).
Lecture 10 We endowed the integers mod n with well-defined
addition and multiplication. We defined the notion of a group, and gave
several examples. Here are some
notes
which will assist you with Sheet 5, Questions 1 and 2 in particular,
and life in general.
Thu Nov 8 Lecture 13 We introduced Euler's φ-function, and proved that it was multiplicative
with respect to coprime arguments (by exploiting the Chinese Remainder
Theorem). We also proved that of f, g are polynomials in K[X]
where K is a field, and g is not the zero polynomial,
then there are q, r in K[X] such
that f = qg + r and deg r < deg g. There were a couple of
glitches. (i) In the proof that deg(f + g) ≤
max {deg f, deg g}, I wrote
(al + bl)Xk+l but the exponent should
be l rather than k + l.
(ii) Also in the calculation of φ(pk), the set being subtracted
should have been { tp | 0 ≤ t < pk-1} (and
not { tp | 0 ≤ t < p}).
Tue Nov 20 Lecture 16 More on linear maps and matrices. Small glitch
in one proof. Fix to follow. How to invert a 2 by 2 real matrix (or indeed, any 2 by 2 matrix
with entries in a field). There was a glitch in the lecture, in (iii)
where we were establishing that if the determinant of a real 2 by 2 matrix
is 0, then it has no inverse. Here is a
fix.
Here is Problem Sheet 9Sheet 8 solutions.
Thu Nov 22 Lecture 17 The area/volume interpretation of determinants for 2 by 2 and 3 by 3
matrices with real entries.
Thu Dec 6 Lecture 20
Each element of S_n is either an even permutation or an odd permutation,
and no permutation is in both categories. Therefore S_n is the union of two cosets of A_n, and
A_n has size n!/2.
Sheet 10 solutions.
Vacation Problem Sheet 11.
Tue Dec 11 Lecture 21 The use of the sign of a permutation to define a determinant of a square matrix
as an alternating sum of monomials.
The email address
[email protected]
is the
official University of Bath format, though once upon a time the shiny new
format was [email protected] --
the one that people actually use
is, of course, entirely different: [email protected] -- as far
as I know, all these incantations work. |
Discrete Mathematics With Combinatorics
9780130457912
ISBN:
0130457914
Edition: 2 Pub Date: 2003 Publisher: Prentice Hall PTR
Summary: As in the first edition, the purpose of this book is to present an extensive range and depth of topics in discrete mathematics and also work in a theme on how to do proofs. Proofs are introduced in the first chapter and continue throughout the book. Most students taking discrete mathematics are mathematics and computer science majors. Although the necessity of learning to do proofs is obvious for mathematics majors, ...it is also critical for computer science students to think logically. Essentially, a logical bug-free computer program is equivalent to a logical proof. Also, it is assumed in this book that it is easier to use (or at least not misuse) an application if one understands why it works. With few exceptions, the book is self-contained. Concepts are developed mathematically before they are seen in an applied context. Additions and alterations in the second edition: More coverage of proofs, especially in Chapter 1. Added computer science applications, such as a greedy algorithm for coloring the nodes of a graph, a recursive algorithm for counting the number of nodes on a binary search tree, or an efficient algorithm for computinga b modnfor very large values ofa, b,andn. An extensive increase in the number of problems in the first eight chapters. More problems are included that involve proofs. Additional material is included on matrices. Inclusion of finite states with output and Turing machines. True-False questions at the end of each chapter. Summary questions at the end of each chapter. A glossary at the end of each chapter. Functions and sequences are introduced earlier (in Chapter 2). Calculus is not required for any of the material in this book. College algebra is adequate for the basic chapters. However, although this book is self-contained, some of the remaining chapters require more mathematical maturity than do the basic chapters, so calculus is recommended more for giving maturity, than for any direct uses. This book is intended for either a one- or two-term course in discrete mathematics. The first eight chapters of this book provide a foundation in discrete mathematics and would be appropriate for a first-level course for freshmen or sophomores. These chapters are essentially independent, so that the instructor can pick the material he/she wishes to cover. The remainder of the book contains appropriate material for a second course in discrete mathematics. These chapters expand concepts introduced earlier and introduce numerous advanced topics. Topics are explored from different points of view to show how they may be used in different settings. The range of topics include: Logic--Including truth tables, propositional logic, predicate calculus, circuits, induction, and proofs. Set Theory--Including cardinality of sets, relations, partially ordered sets, congruence relations, graphs, directed graphs, and functions. Algorithms--Including complexity of algorithms, search and sort algorithms, the Euclidean algorithm, Huffman's algorithm, Prim's algorithms, Warshall's algorithm, the Ford-Fulkerson algorithm, the Floyd-Warshall algorithm, and Dijkstra's algorithms. Graph Theory--Including directed graphs, Euler cycles and paths, Hamiltonian cycles and paths, planar graphs, and weighted graphs. Trees--including binary search trees, weighted trees, tree transversal, Huffman's codes, and spanning trees. Combinatorics--including permutations, combinations, inclusion-exclusion, partitions, generating functions, Catalan numbers, Sterling numbers, Rook Polynomials, derangements, and enumeration of colors. Algebra--Including semigroups, groups, lattices, semilattices, Boolean algebras, rings, fields, integral domains,[ |
Determinants
In this lesson on Determinants, you will learn the basics of a determinant such as the requirement of square matrices and using vertical bars to differentiate them from standard matrices. You will then move onto finding the second order determinant of a 2x2 matrix before moving into finding the third order determinant of a 3x3 matrix. For a 3x3 matrix you first learn the technique called expansion by minors before moving into the diagonal method. Four examples at the end test your new found knowledge.
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Determinants
Third-order determinants can be evaluated by expanding by minors along any row or column of the associated matrix.
These determinants can also be found by calculating products of entries along the diagonals and then adding and subtracting these products.
Determinants
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
Just because something doesn't teach everything about a subject doesn't make it a poor instructional aid. Algebra is more than rote memorization, but having the rules memorized does help a lot in getting to the stage where you truly understand it. Even if you do understand it and could in principle recreate the rules from scratch, recalling them from memory is faster than deriving them each time you need them - and since algebra is pretty much the foundation of all advanced math, you'll be needing them a lot if you want to study math at all. (Though if you end up struggling with the harder topics because you didn't have the rules of algebra appropriately memorized, you might never want to study more of it...)
Algebra is more than rote memorization, but having the rules memorized does help a lot in getting to the stage where you truly understand it.
I disagree. I think that memorizing the rules first, without understanding where they come from, discourages the student from attempting to understand anything to begin with. After all, his goal is to balance an equation, and look, he just balanced it... so what else is there to know ? Thus, the memorization approach creates the impression that math (or whatever subject you're studying) is all about arbitrary rules that make no sense; it's all about "guessing the teacher's password", and that's boring.
Contrast this with the approach of treating an equation like a puzzle. If "2x - 3 = 5", and we want to know what x is, there are many ways to approach the solution. We could ask, "someone did a bunch of stuff to x to get 5, how can we undo it ?", or we could say, "the equation is like a pair of scales that are balanced, so what can we do to get x by itself without unbalancing the scales ?", etc. Some possible partial answers are, "someone took away 3, so let's add it back", or "if we add 3 to both sides, the scales will still be balanced but we'll be one step closer to a solution". But "add 3 to both sides because that's how the game is programmed and you won't get the high score otherwise" isn't much of an answer. High scores don't mean anything, algebra does.
Well, I can't speak for others, but my personal experience with math tends to be that I only start properly learning why something works once I have the rules pretty well memorized. Before that, my working memory is so occupied with trying to just remember how to apply the rules that I don't have the space to remember why they work. Or alternatively, I can learn why the rules work - but in that case I don't have the memory capacity left for remembering how to apply them.
Of course, this is complicated by the fact that during the process of trying to memorize the rules, I often stop to think about why they work in an attempt to rederive them and make sure I'm not misremembering them. So it's not pure rote memorization, like the way it seems to be with DragonBox. But I would still expect that if somebody first learned them as meaningless rules in the game, and was then later taught math and the reasons for the rules, they'd have a good chance of being delighted at discovering where the rules came from, and could spend all of their cognitive capacity on developing an actual understanding.
Fair enough; it's possible that you and I simply think in different ways. I personally find it very difficult to memorize (seemingly) arbitrary rules, and I found it very difficult to un-teach the "guess the teacher's password" mentality to people. But it's quite likely that I'm making an unjustified generalization from a very small number of examples. |
Find a LonetreeSystems of equations and inequalities are examined both graphically and algebraically. This course also touches on complex numbers and DeMoivre's theorem as well as sequences, induction, counting, and probability. As a PhD student, I spent many hours reading complex books and articles and I am confident that I can help your student achieve higher reading comprehension levels |
Matrices
Summary: This chapter covers principles of matrices. After completing this chapter students should be able to: complete matrix operations; solve linear systems using Gauss-Jordan method; Solve linear systems using the matrix inverse method and complete application problems.
Chapter Overview
In this chapter, you will learn to:
Do matrix operations.
Solve linear systems using the Gauss-Jordan method.
Solve linear systems using the matrix inverse method.
Do application problems.
Introduction to Matrices
Section Overview
In this section you will learn to:
Add and subtract matrices.
Multiply a matrix by a scalar.
Multiply two matrices.
A matrix is a rectangular array of numbers. Matrices are useful in organizing and manipulating large amounts of data. In order to get some idea of what matrices are all about, we will look at the following example.
Example 1
Problem 1
Fine Furniture Company makes chairs and tables at its San Jose, Hayward, and Oakland factories. The total production, in hundreds, from the three factories for the years 1994 and 1995 is listed in the table below.
Table 1
1994
1995
Chairs
Tables
Chairs
Tables
San Jose
30
18
36
20
Hayward
20
12
24
18
Oakland
16
10
20
12
Represent the production for the years 1994 and 1995 as the matrices A and B.
Find the difference in sales between the years 1994 and 1995.
The company predicts that in the year 2000 the production at these factories will double that of the year 1994. What will the production be for the year 2000?
Before we go any further, we need to familiarize ourselves with some terms that are associated with matrices. The numbers in a matrix are called the entries or the elements of a matrix. Whenever we talk about a matrix, we need to know the size or the dimension of the matrix. The dimension of a matrix is the number of rows and columns it has. When we say a matrix is a 3 by 4 matrix, we are saying that it has 3 rows and 4 columns. The rows are always mentioned first and the columns second. This means that a
3×43×4 size 12{3 times 4} {} matrix does not have the same dimension as a
4×34×3 size 12{4 times 3} {} matrix. A matrix that has the same number of rows as columns is called a square matrix. A matrix with all entries zero is called a zero matrix. A square matrix with 1's along the main diagonal and zeros everywhere else, is called an identity matrix. When a square matrix is multiplied by an identity matrix of same size, the matrix remains the same. A matrix with only one row is called a row matrix or a row vector, and a matrix with only one column is called a column matrix or a column vector. Two matrices are equal if they have the same size and the corresponding entries are equal.
Matrix Addition and Subtraction
If two matrices have the same size, they can be added or subtracted. The operations are performed on corresponding entries.
To multiply a matrix by another is not as easy as the addition, subtraction, or scalar multiplication of matrices. Because of its wide use in application problems, it is important that we learn it well. Therefore, we will try to learn the process in a step by step manner. We first begin by finding a product of a row matrix and a column matrix.
Solution
We already know how to multiply a row matrix by a column matrix. To find the product
ABAB size 12{ ital "AB"} {}, in this example, we will be multiplying the row matrix
AA size 12{A} {} to both the first and second columns of matrix
BB size 12{B} {}, resulting in a
1×21×2 size 12{1 times 2} {} matrix.
We have just multiplied a
1×31×3 size 12{1 times 3} {} matrix by a matrix whose size is
3×23×2 size 12{3 times 2} {}. So unlike addition and subtraction, it is possible to multiply two matrices with different dimensions as long as the number of entries in the rows of the first matrix are the same as the number of entries in columns of the second matrix.
Solution
This time we are multiplying two rows of the matrix
AA size 12{A} {} with two columns of the matrix
BB size 12{B} {}. Since the number of entries in each row of
AA size 12{A} {} are the same as the number of entries in each column of
BB size 12{B} {}, the product is possible. We do exactly what we did in Example 6. The only difference is that the matrix
AA size 12{A} {} has one more row.
We multiply the first row of the matrix
AA size 12{A} {} with the two columns of
BB size 12{B} {}, one at a time, and then repeat the process with the second row of
AA size 12{A} {}. We get
The product
FEFE size 12{ ital "FE"} {} is not possible because the matrix
FF size 12{F} {} has two entries in each row, while the matrix
EE size 12{E} {} has three entries in each column. In other words, the matrix
FF size 12{F} {} has two columns, while the matrix
EE size 12{E} {} has three rows.
In order for product
ABAB size 12{ ital "AB"} {} to exist, the number of columns of
AA size 12{A} {}, must equal the number of rows of
BB size 12{B} {}. If matrix
AA size 12{A} {} is of dimension
m×nm×n size 12{m times n} {} and
BB size 12{B} {} of dimension
n×pn×p size 12{n times p} {}, the product will have the dimension
m×pm×p size 12{m times p} {}. Furthermore, matrix multiplication is not commutative.
In this chapter, we will be using matrices to solve linear systems. In Section 7, we will be asked to express linear systems as the matrix equation AX=BAX=B size 12{ ital "AX"=B} {}, where
AA size 12{A} {},
XX size 12{X} {}, and
BB size 12{B} {} are matrices. The matrix
AA size 12{A} {} is called the coefficient matrix.
Systems of Linear Equations; Gauss-Jordan Method
In this section, we learn to solve systems of linear equations using a process called the Gauss-Jordan method. The process begins by first expressing the system as a matrix, and then reducing it to an equivalent system by simple row operations. The process is continued until the solution is obvious from the matrix. The matrix that represents the system is called the augmented matrix, and the arithmetic manipulation that is used to move from a system to a reduced equivalent system is called a row operation.
Example 12
Problem 1
Write the following system as an augmented matrix.
2x+3y−4z=52x+3y−4z=5 size 12{2x+3y - 4z=5} {}
(30)
3x+4y−5z=−63x+4y−5z=−6 size 12{3x+4y - 5z= - 6} {}
(31)
4x+5y−6z=74x+5y−6z=7 size 12{4x+5y - 6z=7} {}
(32)
Solution
We express the above information in matrix form. Since a system is entirely determined by its coefficient matrix and by its matrix of constant terms, the augmented matrix will include only the coefficient matrix and the constant matrix. So the augmented matrix we get is as follows:
In the Section 2, we expressed the system of equations as
AX=BAX=B size 12{ ital "AX"=B} {}, where
AA size 12{A} {} represented the coefficient matrix, and
BB size 12{B} {} the matrix of constant terms. As an augmented matrix, we write the matrix as
A∣BA∣B size 12{ left [A \lline B right ]} {}. It is clear that all of the information is maintained in this matrix form, and only the letters
xx size 12{x} {},
yy size 12{y} {} and
zz size 12{z} {} are missing. A student may choose to write
xx size 12{x} {},
yy size 12{y} {} and
zz size 12{z} {} on top of the first three columns to help ease the transition.
Example 13
Problem 1
For the following augmented matrix, write the system of equations it represents.
Once a system is expressed as an augmented matrix, the Gauss-Jordan method reduces the system into a series of equivalent systems by employing the row operations. This row reduction continues until the system is expressed in what is called the reduced row echelon form. The reduced row echelon form of the coefficient matrix has 1's along the main diagonal and zeros elsewhere. The solution is readily obtained from this form.
The method is not much different form the algebraic operations we employed in the elimination method in the first chapter. The basic difference is that it is algorithmic in nature, and, therefore, can easily be programmed on a computer.
We will next solve a system of two equations with two unknowns, using the elimination method, and then show that the method is analogous to the Gauss-Jordan method.
Example 14
Problem 1
Solve the following system by the elimination method.
x+3y=7x+3y=7 size 12{x+3y=7} {}
(38)
3x+4y=113x+4y=11 size 12{3x+4y="11"} {}
(39)
Solution
We multiply the first equation by – 3, and add it to the second equation.
Row Operations
One can easily see that these three row operation may make the system look different, but they do not change the solution of the system.
The first row operation states that if any two rows of a system are interchanged, the new system obtained has the same solution as the old one. Let us look at an example in two equations with two unknowns. Consider the system
x+3y=7x+3y=7 size 12{x+3y=7} {}
(53)
3x+4y=113x+4y=11 size 12{3x+4y="11"} {}
(54)
We interchange the rows, and we get,
3x+4y=113x+4y=11 size 12{3x+4y="11"} {}
(55)
x+3y=7x+3y=7 size 12{x+3y=7} {}
(56)
Clearly, this system has the same solution as the one above.
The second operation states that if a row is multiplied by any non-zero constant, the new system obtained has the same solution as the old one. Consider the above system again,
x+3y=7x+3y=7 size 12{x+3y=7} {}
(57)
3x+4y=113x+4y=11 size 12{3x+4y="11"} {}
(58)
We multiply the first row by –3, we get,
-3x-9y=-21-3x-9y=-21
(59)
3x+4y=113x+4y=11 size 12{3x+4y="11"} {}
(60)
Again, it is obvious that this new system has the same solution as the original.
The third row operation states that any constant multiple of one row added to another preserves the solution. Consider our system,
x+3y=7x+3y=7 size 12{x+3y=7} {}
(61)
3x+4y=113x+4y=11 size 12{3x+4y="11"} {}
(62)
If we multiply the first row by – 3, and add it to the second row, we get,
x+3y=7x+3y=7 size 12{x+3y=7} {}
(63)
−5y=−10−5y=−10 size 12{ - 5y= - "10"} {}
(64)
And once again, the same solution is maintained.
Now that we understand how the three row operations work, it is time to introduce the Gauss-Jordan method to solve systems of linear equations.
As mentioned earlier, the Gauss-Jordan method starts out with an augmented matrix, and by a series of row operations ends up with a matrix that is in the reduced row echelon form. A matrix is in the reduced row echelon form if the first nonzero entry in each row is a 1, and the columns containing these 1's have all other entries as zeros. The reduced row echelon form also requires that the leading entry in each row be to the right of the leading entry in the row above it, and the rows containing all zeros be moved down to the bottom.
We state the Gauss-Jordan method as follows.
Gauss-Jordan Method
Write the augmented matrix.
Interchange rows if necessary to obtain a non-zero number in the first row, first column.
Use a row operation to make the entry in the first row, first column, a 1.
Use row operations to make all other entries as zeros in column one.
Interchange rows if necessary to obtain a nonzero number in the second row, second column. Use a row operation to make this entry 1. Use row operations to make all other entries as zeros in column two.
Repeat step 5 for row 3, column 3. Continue moving along the main diagonal until you reach the last row, or until the number is zero. The final matrix is called the reduced row-echelon form.
We want a 1 in row one, column one. This can be obtained by dividing the first row by 2, or interchanging the second row with the first. Interchanging the rows is a better choice because that way we avoid fractions.
So far we have made a 1 in the left corner and all other entries zeros in that column. Now we move to the next diagonal entry, row 2, column 2. We need to make this entry(–3) a 1 and make all other entries in this column zeros. To make row 2, column 2 entry a 1, we divide the entire second row by –3.
Before we leave this section, we mention some terms we may need in the fourth chapter. The process of obtaining a 1 in a location, and then making all other entries zeros in that column, is called pivoting. The number that is made a 1 is called the pivot element, and the row that contains the pivot element is called the pivot row. We often multiply the pivot row by a number and add it to another row to obtain a zero in the latter. The row to which a multiple of pivot row is added is called the target row.
Systems of Linear Equations – Special Cases
Section Overview
In this section you will learn to:
Determine the linear systems that have no solution.
Solve the linear systems that have infinitely many solutions.
If we consider the intersection of two lines in a plane, three things can happen.
The lines intersect in exactly one point. This is called an independent system.
The lines are parallel, so they do not intersect. This is called an inconsistent system.
The lines coincide, so they intersect at infinitely many points. This is a dependent system.
The figures below shows all three cases.
Figure 1
Every system of equations has either one solution, no solution, or infinitely many solutions.
In the Section 4, we used the Gauss-Jordan method to solve systems that had exactly one solution. In this section, we will determine the systems that have no solution, and solve the systems that have infinitely many solutions.
Example 19
Problem 1
Solve the following system of equations.
x+y=7x+y=7 size 12{x+y=7} {}
(76)
x+y=9x+y=9 size 12{x+y=9} {}
(77)
Solution
Let us use the Gauss-Jordan method to solve this system. The augmented matrix is as follows.
Example 21
Problem 1
Solution
The problem clearly asks for the intersection of two lines that are the same; that is, the lines coincide. This means the lines intersect at an infinite number of points.
A few intersection points are listed as follows: (3, 4), (5, 2), (–1, 8), (–6, 13) etc. However, when a system has an infinite number of solutions, the solution is often expressed in the parametric form. This can be accomplished by assigning an arbitrary constant,
tt size 12{t} {}, to one of the variables, and then solving for the remaining variables. Therefore, if we let
y=ty=t size 12{y=t} {}, then
x=7−tx=7−t size 12{x=7 - t} {}. Or we can say all ordered pairs of the form (
7−t7−t size 12{7 - t} {},
tt size 12{t} {}) satisfy the given system of equations.
Alternatively, while solving the Gauss-Jordan method, we will get the reduced row-echelon form given below.
The row of all zeros, can simply be discarded in a manner that it never existed. This leaves us with only one equation but two variables. And whenever there are more variables than the equations, the solution must be expressed in terms of an arbitrary constant, as above. That is,
x=7−tx=7−t size 12{x=7 - t} {},
y=ty=t size 12{y=t} {}.
Since the last equation dropped out, we are left with two equations and three variables. This means the system has infinite number of solutions. We express those solutions in the parametric form by letting the last variable
zz size 12{z} {} equal the parameter
tt size 12{t} {}.
The reader should note that particular solutions to the system can be obtained by assigning values to the parameter
tt size 12{t} {}. For example, if we let
t=2t=2 size 12{t=2} {}, we have the solution (5, –5, 2).
This time the last two equations drop out, and we are left with one equation and three variables. Again, there are infinite number of solutions. But this time the answer must be expressed in terms of two arbitrary constants.
If any row of the reduced row-echelon form of the matrix gives a false statement such as 0 = 1, the system is inconsistent and has no solution.
If the reduced row echelon form has fewer equations than the variables and the system is consistent, then the system has an infinite number of solutions. Remember the rows that contain all zeros are dropped.
If a system has an infinite number of solutions, the solution must be expressed in the parametric form.
The number of arbitrary parameters equals the number of variables minus the number of equations.
Inverse Matrices
Section Overview
In this section you will learn to:
Find the inverse of a matrix, if it exists.
Use inverses to solve linear systems.
In this section, we will learn to find the inverse of a matrix, if it exists. Later, we will use matrix inverses to solve linear systems.
As we look at the two augmented matrices, we notice that the coefficient matrix for both the matrices is the same. Which implies the row operations of the Gauss-Jordan method will also be the same. A great deal of work can be saved if the two right hand columns are grouped together to form one augmented matrix as below.
Now that we know how to find the inverse of a matrix, we will use inverses to solve systems of equations. The method is analogous to solving a simple equation like the one below.
23x=423x=4 size 12{ { {2} over {3} } x=4} {}
(122)
Example 28
Problem 1
Solve the following equation .
23x=423x=4 size 12{ { {2} over {3} } x=4} {}
(123)
Solution
To solve the above equation, we multiply both sides of the equation by the multiplicative inverse of
2323 size 12{ { {2} over {3} } } {} which happens to be
3232 size 12{ { {3} over {2} } } {} . We get
To solve a linear system, we first write the system in the matrix equation
AX=BAX=B size 12{ ital "AX"=B} {}, where
AA size 12{A} {} is the coefficient matrix,
XX size 12{X} {} the matrix of variables, and
BB size 12{B} {} the matrix of constant terms. We then multiply both sides of this equation by the multiplicative inverse of the matrix
AA size 12{A} {}.
Consider the following example.
Example 29
Problem 1
Solve the following system
3x+y=33x+y=3 size 12{3x+y=3} {}
(126)
5x+2y=45x+2y=4 size 12{5x+2y=4} {}
(127)
Solution
To solve the above equation, first we express the system as
AX=BAX=B size 12{ ital "AX"=B} {}
(128)
where
AA size 12{A} {} is the coefficient matrix, and
BB size 12{B} {} is the matrix of constant terms. We get
Once again, we remind the reader that not every system of equations can be solved by the matrix inverse method. Although the Gauss-Jordan method works for every situation, the matrix inverse method works only in cases where the inverse of the square matrix exists. In such cases the system has a unique solution.
Application of Matrices in Cryptography
In this section, we see a use of matrices in encoding and decoding secret messages. There are many techniques used, but we will use a method that first converts the secret message into a string of numbers by arbitrarily assigning a number to each letter of the message. Next we convert this string of numbers into a new set of numbers by multiplying the string by a square matrix of our choice that has an inverse. This new set of numbers represents the coded message. To decode the message, we take the string of coded numbers and multiply it by the inverse of the matrix to get the original string of numbers. Finally, by associating the numbers with their corresponding letters, we obtain the original message.
In this section, we will use the correspondence where the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27, and all punctuation is ignored.
Example 31
Problem 1
Solution
We divide the letters of the message into groups of two.
AT TA CK –N OW
We assign the numbers to these letters from the above table, and convert each pair of numbers into
2×12×1 size 12{2 times 1} {} matrices. In the case where a single letter is left over on the end, a space is added to make it into a pair.
TO ENCODE A MESSAGE
Convert each group into a string of numbers by assigning a number to each letter of the message. Remember to assign letters to blank spaces.
Convert each group of numbers into column matrices.
Convert these column matrices into a new set of column matrices by multiplying them with a compatible square matrix of your choice that has an inverse. This new set of numbers or matrices represents the coded message.
TO DECODE A MESSAGE
Take the string of coded numbers and multiply it by the inverse of the matrix that was used to encode the message.
Associate the numbers with their corresponding letters.
Applications – Leontief Models
In the 1930's, Wassily Leontief used matrices to model economic systems. His models, often referred to as the input-output models, divide the economy into sectors where each sector produces goods and services not only for itself but also for other sectors. These sectors are dependent on each other and the total input always equals the total output. In 1973, he won the Nobel Prize in Economics for his work in this field. In this section we look at both the closed and the open models that he developed.
The Closed Model
As an example of the closed model, we look at a very simple economy, where there are only three sectors: food, shelter, and clothing.
Example 36
Problem 1
We assume that in a village there is a farmer, carpenter, and a tailor, who provide the three essential goods: food, shelter, and clothing. Suppose the farmer himself consumes 40% of the food he produces, and gives 40% to the carpenter, and 20% to the tailor. Thirty percent of the carpenter's production is consumed by himself, 40% by the farmer, and 30% by the carpenter. Fifty percent of the tailor's production is used by himself, 30% by the farmer, and 20% by the tailor. Write the matrix that describes this closed model.
This matrix is called the input-output matrix. It is important that we read the matrix correctly. For example the entry
A23A23 size 12{A rSub { size 8{"23"} } } {}, the entry in row 2 and column 3, represents the following.
A23=20%A23=20% size 12{A rSub { size 8{"23"} } ="20"%} {} of the tailor's production is used by the carpenter.
A33=50%A33=50% size 12{A rSub { size 8{"33"} } ="50"%} {}of the tailor's production is used by the tailor.
As we said earlier, in this model input must equal output. That is, the amount paid by each equals the amount received by each.
Let us say the farmer gets paid
xx size 12{x} {} dollars. Let us now look at the farmer's expenses. The farmer uses up 40% of his own production, that is, of the
xx size 12{x} {} dollars he gets paid, he pays himself
.40x.40x size 12{ "." "40"x} {} dollars, he pays
.40y.40y size 12{ "." "40"y} {} dollars to the carpenter, and
.30z.30z size 12{ "." "30"z} {} to the tailor. Since the expenses equal the wages, we get the following equation.
Note:
The use of a calculator in solving these problems is strongly recommended. Although we at De Anza College use TI-85 calculators, any calculator that handles matrices will do.
The Open Model
The open model is more realistic, as it deals with the economy where sectors of the economy not only satisfy each others needs, but they also satisfy some outside demands. In this case, the outside demands are put on by the consumer. But the basic assumption is still the same; that is, whatever is produced is consumed.
Let us again look at a very simple scenario. Suppose the economy consists of three people, the farmer
FF size 12{F} {}, the carpenter
CC size 12{C} {}, and the tailor
TT size 12{T} {}. A part of the farmer's production is used by all three, and the rest is used by the consumer. In the same manner, a part of the carpenter's and the tailor's production is used by all three, and rest is used by the consumer.
Let us assume that whatever the farmer produces, 20% is used by him, 15% by the carpenter, 10% by the tailor, and the consumer uses the other 40 billion dollars worth of the food. Ten percent of the carpenter's production is used by him, 25% by the farmer, 5% by the tailor, and 50 billion dollars worth by the consumer. Fifteen percent of the clothing is used by the tailor, 10% by the farmer, 5% by the carpenter, and the remaining 60 billion dollars worth by the consumer. We write the internal consumption in the following table, and express the demand as the matrix D.
Table 4
FF size 12{F} {} produces
CC size 12{C} {} produces
TT size 12{T} {} produces
FF size 12{F} {} uses
.20
.25
.10
CC size 12{C} {} uses
.15
.10
.05
TT size 12{T} {} uses
.10
.05
.15
The consumer demand for each industry in billions of dollars is given below.
In the closed model, our equation was
X=AXX=AX size 12{X = ital "AX"} {}, that is, the total input equals the total output. This time our equation is similar with the exception of the demand by the consumer.
So our equation for the open model should be
X=AX+DX=AX+D size 12{X= ital "AX"+D} {}, where
DD size 12{D} {} represents the demand matrix. We express it as follows:
We will do one more problem like the one above, except this time we give the amount of internal and external consumption in dollars and ask for the proportion of the amounts consumed by each of the industries. In other words, we ask for the matrix A.
Example 39
Problem 1
Suppose an economy consists of three industries
FF size 12{F} {},
CC size 12{C} {}, and
TT size 12{T} {}. Again, each of the industries produces for internal consumption among themselves, as well as, the external demand by the consumer. The following table gives information about the use of each industry's production in dollars.
Table 5
FF size 12{F} {}
CC size 12{C} {}
TT size 12{T} {}
Demand
Total
FF size 12{F} {}
40
50
60
100
250
CC size 12{C} {}
30
40
40
110
220
TT size 12{T} {}
20
30
30
120
200
The first row says that of the $250 dollars worth of production by the industry
FF size 12{F} {}, $40 is used by
FF size 12{F} {}, $50 is used by
CC size 12{C} {}, $60 is used by
TT size 12{T} {}, and the remainder of $100 is used by the consumer. The other rows are described in a similar manner.
Find the proportion of the amounts consumed by each of the industries. In other words, find the matrix A.
Once again, the total input equals the total output.
Solution
We are being asked to determine the following:
How much of the production of each of the three industries,
FF size 12{F} {},
CC size 12{C} {}, and
TT size 12{T} {} is required to produce one unit of
FF size 12{F} {}? In the same way, how much of the production of each of the three industries,
FF size 12{F} {},
CC size 12{C} {}, and
TT size 12{T} {} is required to produce one unit of
CC size 12{C} {}? And finally, how much of the production of each of the three industries,
FF size 12{F} {},
CC size 12{C} {}, and
TT size 12{T} {} is required to produce one unit of
TT size 12{T} {}?
Since we are looking for proportions, we need to divide the production of each industry by the total production for each industry |
0471510025
9780471510024 rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry. «Show less... Show more»
Rent Elementary Geometry 3rd Edition today, or search our site for other Frisk |
- Get to grips with converting your mathematics teaching over to Moodle - Engage and motivate your students with exciting, interactive, and engaging online math courses with Moodle, which include mathematical notation, graphs, images, video, audio, and more - Integrate multimedia elements in math courses to make learning math interactive and fun - Inspiring, realistic examples and interactive assessment exercises to give you ideas for your own Moodle math courses
In Detail Moodle is a popular e-learning platform that is making inroads into all areas of the curriculum. Using moodle helps you to develop exciting, interactive, and engaging online math courses. But teaching math requires use of graphs, equations, special notation, and other features that are not built into Moodle. Using Moodle to teach Mathematics presents its own challenges.
The book will show you how to set-up a Moodle course to support the teaching of mathematics. It will also help you to carefully explore the Moodle plugins that allow the handling of equations and enable other frequently used mathematical activities.
Taking a practical approach, this book will introduce you to the concepts of converting mathematics teaching over to Moodle. It provides you with everything you need to include mathematical notation, graphs, images, video, audio, and more in your Moodle courses. By following the practical examples in this book, you can create feature-rich quizzes that are automatically marked, use tools to monitor student progress, employ modules and plugins allowing students to explore mathematical concepts. You'll also learn the integration of presentations, interactive math elements, SCORM, and Flash objects into Moodle. It will take you through these elements in detail and help you learn how to create, edit, and integrate them into Moodle.
Soon you will develop your own exciting, interactive, and engaging online math courses with ease.
What you will learn from this book? - Convert mathematics teaching over to Moodle - Enhance your course with interactive graphs, images, videos, and audio - Integrate interactive presentations and explore different ways to include them in your course - Create your own SCORM activities using both free and commercial tools - Add rich animation and fun games by incorporating Flash games and activities for engaging your students - Build feature-rich quizzes and set online assignments - Monitor student progress and assess your teaching success - Configure Moodle to display the complete set of mathematical symbols and objects
Approach The book presents the reader with clear instructions for setting up specific activities, based around an example maths course (Pythagorean Theorem) with plenty of examples and screenshots. No Moodle experience is required to use the book, but the book will focus only on activities and modules relevant to teaching mathematics. We will assume that the reader has access to a working installation of Moodle. The activities will be appropriate for teaching math in high schools and universities.
Who this book is written for? The book is aimed at math teachers who want to use Moodle to deliver or support their teaching. The book will also be useful for teachers of "mathematical sciences", or courses with a significant mathematical content that will benefit from the use of some of the tools explored in the book. No Moodle experience is required to use the book.
Moodle 1.9 Math |
Mathematics
Page Content
MATH 100. Basic College Mathematics (3; F, S)
Three hours per week. This
course may not be used to satisfy the University's Core mathematics
requirement. Students may not enroll in this course if they have
satisfactorily completed a higher numbered MATH course. An overview of basic
algebraic and geometric skills. This course is designed for students who lack
the needed foundation in college level mathematics. A graphing calculator is
required.
MATH 104. College Algebra (3; F, S) Three hours per week.
Prerequisite: MATH 100. This course may
not be used to satisfy the University's Core mathematics requirement.
Qualitative and quantitative aspects of linear, exponential, rational, and
polynomial functions are explored using a problem solving approach. Basic
modeling techniques, communication, and the use of technology is emphasized. A
graphing calculator is required.
MATH 110. The Mathematics of Motion & Change (3; F, S) Three hours per week. Prerequisite: MATH 104. A study of the mathematics of
growth, motion and change. A review of algebraic, exponential, and trigonometric
functions. This course is designed as a terminal course or to prepare students
for the sequence of calculus courses. A graphing calculator is required.
MATH 112. Modern Applications of Mathematics (3; F, S)
Three hours per week. Prerequisite: MATH 104. Calculus concepts as
applied to real-world problems. Topics include applications of polynomial and
exponential functions and the mathematics of finance. A graphing calculator is
required.
MATH 140. Calculus I (4; F, S) Four hours per week.
Prerequisite: A "C" or better in MATH 110. Rates of change, polynomial and
exponential functions, models of growth. Differential calculus and its
applications. Simple differential equations and initial value problems. A
graphing calculator is required.
MATH 141. Calculus II (4; F, S) Four hours per week.
Prerequisite: A "C" or better in MATH 140. The definite integral, the
Fundamental Theorem of Calculus, integral calculus and its applications. An
introduction to series including Taylor series and its convergence. A graphing
calculator is required.
MATH 150. Introduction to Discrete Structures (3; S) Three hours per week. Prerequisite: A "C" or better in one of MATH 110, MATH
112 or MATH 140. An introduction to the mathematics of computing. Problem
solving techniques are stressed along with an algorithmic approach. Topics
include representation of numbers, sets and set operations, functions and
relations, arrays and matrices, Boolean algebra, propositional logic, big O and
directed and undirected graphs.
MATH 199. Special Topics (var. 1-4; AR) May be repeated
for credit when topic changes. Selected topics of student interest and
mathematical significance will be treated.
MATH 206. Statistical Methods in Science (4; S) Four
hours per week. Prerequisite: A "C" or better in MATH 140. Credit cannot be awarded for both MATH 205
and MATH 206. Concepts of probability, distributions of random variables,
estimation, hypothesis testing, regression, ANOVA, design of experiments,
testing of assumptions, scientific sampling and use of statistical software.
Many examples will use real data from scientific research. A graphing calculator
is required.
MATH 220WI. Mathematics & Reasoning (3; S) Three
hours per week. Prerequisite: ENGL 103 and a "C" or better in MATH 141.
Fundamentals of mathematical logic, introduction to set theory, methods of proof
and mathematical writing.
MATH 306. Regression & Analysis of Variance Techniques
(3) Three hours per week. Prerequisites: A "C" or better in MATH
141, and a "C" or better in either MATH 205 or MATH 305. Theory of least
squares, simple linear and multiple regression, regression diagnostics, analysis
of variance, applications of techniques to real data and use of statistical
packages.
MATH 307. College Geometry (3) Three hours per week.
Prerequisite: A "C" or better in MATH 141. A critical study of deductive
reasoning used in Euclid's geometry including the parallel postulate and its
relation to non-Euclidean geometries.
MATH / PHIL 330. Symbolic Logic (3) Three hours per week.
A study of modern formal logic, including both sentential logic and predicate
logic. This course will improve students' abilities to reason effectively.
Includes a review of topics such as proof, validity, and the structure of
deductive reasoning.
MATH 351. Applied Mathematics (3; F) Three hours per
week. Prerequisite: A "C" or better in both MATH 300 and MATH 331. Advanced
calculus and differential equations methods for analyzing problems in the
physical and applied sciences. Calculus topics include potentials, Green's
Theorem, Stokes' Theorem, and the Divergence Theorem. Differential equations
topics include series solutions, special functions, and orthogonal
functions.
MATH 354. Introduction to Partial Differential Equations and Modeling
(3; S) Three hours per week. Prerequisite: A "C" or better in both
MATH 300 and MATH 331. Modeling problems in the physical and applied sciences
with partial differential equations, including the heat, potential, and wave
equations. Solution methods for initial value and boundary value problems
including separation of variables, Fourier analysis, and the method of
characteristics.
MATH 400SI. History of Mathematics (3) Three hours per
week. Prerequisite: A "C" or better in MATH 220WI and junior or senior status.
This course may not be used to satisfy
the University's Core mathematics requirement. A study of the history of
mathematics. Students will complete and present a research paper. Students will
gain experience in professional speaking.
MATH 411. Introduction to Real Analysis (3) Three hours
per week. Prerequisite: A "C" or better in both MATH 220WI and MATH 300.
Foundations of real analysis including sequences and series, limits, continuity,
and differentiability. Emphasis on the rigorous formulation and writing of
proofs.
MATH 412. Introduction to Complex Variables (3) Three
hours per week. Prerequisite: A "C" or better in both MATH 220WI and MATH 300.
Algebra of complex numbers, analytic functions, elementary functions, line and
contour integrals, series, residues, poles and applications.
MATH 423. Algebraic Structures (3) Three hours per week.
Prerequisite: A "C" or better in MATH 220WI. An overview of groups, rings,
fields and integral domains. Applications of abstract algebra.
MATH 440. Special Topics (var. 1-3; AR) Prerequisite: A
"C" or better in MATH 220WI or consent of the instructor. May be repeated for
credit when topic changes. Selected topics of student interest and mathematical
significance will be treated.
MATH 501. Introduction to Analysis (3) Three hours per
week. A study of real numbers and the important theorems of differential and
integral calculus. Proofs are emphasized, and a deeper understanding of calculus
is stressed. Attention is paid to calculus reform and the integrated use of
technology.
MATH 502. Survey of Geometries (3) Three hours per week.
An examination of Euclidean and non-Euclidean geometries. Transformational and
finite geometries.
MATH 503. Probability & Statistics (3) Three hours
per week. Probability theory and its role in decision-making, discrete and
continuous random variables, hypothesis testing, estimation, simple linear
regression, analysis of variance and some nonparametric tests. Attention is paid
to statistics reform and the integrated use of technology.
MATH 504. Special Topics (3; AR) Three hours per week.
May be repeated for credit when topic changes. Course content will vary
depending on needs and interests of students.
MATH 507. Number Theory (3) Three hours per week. An
introduction to classical number theory. Topics include modular arithmetic, the
Chinese Remainder Theorem, primes and primality testing, Diophantine equations,
multiplicative functions and continued fractions.
MATH 510. Seminar in the History of Mathematics (3) Three hours per week. Important episodes, problems and discoveries in
mathematics, with emphasis on the historical and social contexts in which they
occurred.
MATH 515. Combinatorics (3) Three hours per week. A
survey of the essential techniques of combinatorics. Applications motivated by
the fundamental problems of existence, enumeration and optimization.
MATH 520. Linear Algebra (3) Three hours per week.
Applications of concepts in linear algebra to problems in mathematical modeling.
Linear systems, vector spaces and linear transformations. Special attention will
be paid to pedagogical considerations.
MATH 531. Theory of Ordinary Differential Equations (3) Three hours per week. Existence and uniqueness theorems. Qualitative and
analytic study of ordinary differential equations, including a study of first
and second order equations, first order systems and qualitative analysis of
linear and nonlinear systems. Modeling of real world phenomena with ordinary
differential equations.
MATH 600. Thesis Seminar (1-3) One to three hours per
week. Research guidance. May be repeated for credit up to a total of three
semester hours.
MATH 699. Thesis Preparation and Research (1) Master of
Arts in Mathematics students who have not completed their thesis and are not
enrolled in any other graduate course must enroll in MATH 699 each fall and
spring semester until final approval of their thesis. This course is Pass/Fail
and does not count towards any graduate degree. |
The CCSS for Mathematical Practice reflect "how" students should interact with math content to master essential skills and their underlying concepts. Math Solutions Common Core courses are specifically designed to align what teachers already know with what they need to know about developing expertise in the "processes and proficiencies" outlined in the Standards for Mathematical Practice.
Students with a strong start to their mathematics education—one that encourages conceptual understanding, procedural fluency, and computational automaticity—will be better prepared for academic success. Math Solutions helps teachers deepen content understanding, which will allow them to build a strong mathematical foundation for their students.
Math Solutions helps teachers incorporate literature and communication to promote thinking and reasoning and increase their students' problem-solving ability. In addition, real-life scenarios and classroom discussion advance students' understanding and ability to use and apply mathematical concepts in a multitude of contexts.
Within a mathematics class, students exhibit a wide variety of learning styles and instructional needs. Math Solutions helps teachers develop strategies for adapting lessons to facilitate understanding for the diversity of learners in their classroom.
Some students need more support, more time, and specialized instruction to learn. Math Solutions helps teachers provide intervention instruction that meets the needs of these struggling students and helps them succeed.
Math Solutions helps high school classroom teachers understand how students learn mathematics, explores ways to make math accessible for students, and focuses on problem solving in the strands of algebra and geometry. |
Book Description: This manual allows students to use Derive as an investigative tool to explore the concepts behind calculus. Each chapter begins with worked examples, followed by exercises and exploration and discovery problems which encourage students to investigate ideas on their own or in groups. |
Description: This course reviews the fundamentals of elementary and intermediate algebra with applications to business and social science. Topics include: using percents, reading and constructing graphs, Venn diagrams, developing quantitative literacy skills, organizing and analyzing data, counting techniques, and elementary probability. Students are also exposed to using technology as graphical and computational aids to solving problems. This course does not satisfy any requirements for the Interdisciplinary Science major. |
Here is a new site that has some nice dynamic calculus tutorials. and here is a link to another good tutorial site called Visual Calculus TI Calculator Guide.. Here is a great link that lets you look up functions on your calculator in an alphabetical list, and then shows you how to do it... simply Great.
HEY, WAY COOL, FREE SOFTWARE
This link will download the WINPLOT program. The file is a self extracting compressed file, just double click to expand. It will create a new file called winplots that is the execute file you want to run when you run the program. And if that didn't convince you, you can also download a Discrete Math software program that is also a great tool.
Another graphing software program which is FREE is called GraphCalc.
You will find it at
Here is another great interactive Algebra and Geometry software, and it is also free. They call it GEOGEBRA.
Lots of Middle Grade teachers have asked about interesting math games for their students which are both educational and entertaining. This link will download a set of Arcade games including John Conway's Game of Life, the 15 puzzle, ghost mazes, and several others. Here are the links to documents I have written about assorted topics.
And for the stats TEACHER a FREE demo of FATHOM my very favorite software for statistics and probability simulations. Students can order a student version for less than $40.
Here is an index of DISCOVERY UNITS using the GEOMETER'S SKETCHPAD that I have written. Some are about GEOMETRY, and some are about ALGEBRA. I hope to have more added soon, so keep checking back. If you do not have Geometer's Sketchpad you can get a FREE DEMO |
Summary: The Third Edition of the Bittinger Graphs and Models series helps students succeed in algebra by emphasizing a visual understanding of concepts. This latest edition incorporates a new Visualizing the Graph feature that helps students make intuitive connections between graphs and functions without the aid of a graphing calculator.
In addition, students learn problem-solving skills from the Bittinger hallmark five-step problem-solving process coupled with Co...show morennecting the Concepts and Aha! Exercises. As you have come to expect with any Bittinger text, we bring you a complete supplements package including MyMathLab® and the new Instructor and Adjunct Support Manual |
The
matrix
algebra
index
begins
with
applications
and
properties
of
matrices,
works
through
systems
of
linear
equations,
explains
determinants
(including
Cramer's
Rule),
and
finishes
with
lessons
on
eigenvalues
and
eigenvectors.
Each
section
includes
an
introduction
to
the
topic
and
example
problems
as
well
as
notes,
tables
and
diagrams.
This
resource
is
part
of
the
Teaching
Quantitative
Skills
in
...
Full description.
The trigonometry index of S.O.S. Math features a table of trigonometric identities, lessons on functions and formulae, and a section of exercises and solutions. Topics also include the derivatives of trigonometric functions and hyperbolic trigonometry. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. description.
This site features a menu of lessons and reference material on calculus concepts. Featured are several definitions of the derivative, treatments of discontinuity, and discussion of logarithms, integration, and antiderivatives. The sections are presented with clear notation and examples.
Full description.
Brand description.
A handy reference on basic geometry formulas, this site covers distance, area, perimeter, and volume. Simple, straightforward notation, no diagrams or lessons. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. description.
Follow this lesson to review basic exponent manipulation. Worksheets, further lessons, and lists of resources are also available. This resource is part of the Teaching Quantitative Skills in the Geosciences collection. description.
This
general
math
site
offers
reference
material
on
a
host
of
math
topics,
plus
a
math
message
board
and
links
to
relevant
material
online.
The
tables
cover
a
range
of
math
skills,
from
basic
fraction-decimal
conversion
to
the
more
advanced
calculus
and
discrete
math.
The
information
is
presented
in
notation
form,
with
diagrams,
graphs,
and
tables.
The
site
is
available
in
English,
Spanish,
and
French.
...
Full description.
Grade level:
Middle (6-8), High (9-12), College (13-14), College (15-16)
This
excerpt
from
the
CRC
Standard
Mathematical
Tables
and
Formulas
covers
geometry,
excluding
differential
geometry.
It
is
a
reference
for
advanced
students,
and
covers
the
material
in
quick,
condensed
sections
of
notes.
Notes
and
diagrams
are
organized
into
sections
and
subsections,
starting
with
coordinate
systems,
plane
transformations,
lines,
and
polygons
in
two-dimensional
geometry.
The
section
...
Full description.
Grade level:
High (9-12), College (13-14), College (15-16), Graduate / Professional
This
page
emphasizes
the
practical
concepts
of
calculus,
and
is
intended
to
provide
a
new
context
for
the
student
already
familiar
with
much
of
the
material.
The
emphasis
is
on
how
calculus
can
actually
be
used
outside
of
the
classroom,
and
how
the
language
of
calculus
is
important
in
many
other
disciplines.
It
features
articles
for
download,
on
topics
from
exponential
growth
and
decay
to
discontinuities,
...
Full description. |
Numeracy Level 5
Description
This unit seeks to develop the skills of interpretation and communication of graphical information and application of a wide range of numerical skills in everyday and straightforward, generalised contexts.
Recommended entry
While entry is at the discretion of the centre, candidates would normally be expected to have attained Numeracy (Intermediate 1/Level 4).
Assessment
While entry is at the discretion of the centre, candidates would normally be expected to have attained Numeracy (Intermediate 1/Level 4). |
Mathematics
The mathematics curriculum has two primary objectives. The first is to provide students with a thorough grounding in the structure and techniques of the subject area, so that they will be well prepared for future work at the secondary or college level as well as motivated to challenge themselves in these venues. The second is to provide students with an understanding of the utility and power of the subject area and of their current competence and abilities as well as their potential for future development and comprehension.
Towards these objectives, the Mathematics Department faculty promote critical thinking and problem solving skills that will enable students to find success in applying their knowledge of mathematics to other fields. In addition, the faculty develop the skills necessary for students to effectively utilize technology as a mathematical tool for exploration and analysis. Finally, the faculty nurture an appreciation for mathematics as an exact science and of the role it plays in the fields of physical science, art, philosophy, engineering, architecture, and industry.
Mathematics Curriculum Overview
In order to graduate from Avon Old Farms School, students must complete at least three mathematics courses: Algebra 1, Geometry, and Algebra 2 with Trigonometry. Upon completion of Algebra 2 with Trigonometry, students are encouraged to enroll in Advanced Mathematics, Precalculus, or Probability and Statistics. After successfully completing Precalculus, students may elect to take Honors Calculus, Advanced Placement Calculus AB, Advanced Placement Calculus BC, or Advanced Placement Probability and Statistics.
Algebra 1
Algebra 1 introduces the student to fundamental operations using signed numbers and their elementary applications. The goal of Algebra 1 is to develop fluency in working with expressions, equations and variables. Students will extend their experiences with tables, graphs, and learn to solve linear equations, inequalities and systems of linear equations. Students will generate equivalent expressions and begin to apply formulas to methodically solve questions involving motion, speed and distance. Students will simplify polynomials and begin to study and apply strategies to solve quadratic relationships.
Students will use technology to learn, investigate, and develop strategies for analyzing complex situations and mathematical relationships. Topics covered in the course include grouping techniques, exponents, algebraic fractions, linear and quadratic equations, radicals, graphing, inequalities, and the solution of verbal problems.
Algebra 1 Honors
This course is designed for students who have demonstrated a strong ability in previous mathematics courses and who wish to pursue upper-level mathematics courses throughout their academic career. In addition to the topics covered in the regular Algebra 1 course, the honors section studies mathematical modeling, trigonometry, and calculator programming.
Algebra 2 with Trigonometry
This course is a more intensive and extensive study of topics introduced in Algebra 1. The primary objective of the Algebra 2 curriculum is to prepare students for Precalculus or Precalculus Honors. The course is designed to prepare students for college level mathematics and is beneficial for those who will pursue further study in mathematics or related fields. Extensive work is included with equalities, inequalities, absolute value, fractional and negative exponents, radicals, systems of quadratics, logarithms and trigonometric properties. The content of the course is organized around families of functions, including linear, quadratic, exponential, logarithmic, radical and rational functions. Students will learn to represent functions in multiple ways, including verbal descriptions, equations, tables, and graphs. Students will also learn to model real-world situations using functions. To help students prepare for standardized tests, this course provides instruction and practice in a variety of formats. Graphing calculator skills will be taught and used extensively in this course. Throughout this course, students will develop learning strategies, critical thinking skills, and problem solving techniques to prepare for future math courses and college entrance exams.
Algebra 2 with Trigonometry Honors
This course is an extensive, fast-moving study of the fundamental principles of algebra, trigonometry, probability, and statistics to prepare students for Precalculus. Students who earn a high "B" range grade or better in this class usually pursue Honors Precalculus the following year. Topics covered include linear equations, functions, polynomials, complex numbers, quadratic equations, and functions. The honors class will also complete chapters on analytic geometry, exponential functions, trigonometry, sequences, series, and probability. Students completing this class in good standing are prepared to study pre-calculus.
Algebra is the language of calculus. Understanding this, there will be special emphasis early in the year on developing a solid working understanding of the algebraic skills and procedures necessary for success in higher level math courses. Students will learn to define the major concepts in a second year algebra course including polynomials, rational expressions, radical expressions, and complex numbers and then learn how to simplify, add, subtract, multiply and divide these expressions. Other major themes include: solving various types of equations and inequalities, factoring, understanding the concept of a function, and graphing functions on the coordinate plane. Linear and quadratic functions are studied in great detail. Later in the year, students will be introduced to higher degree polynomial functions and associated theorems. Students are introduced to conic sections, exponents and logarithms, right triangle and circular trigonometry, and, if time permits, sequences and series.
Geometry
Geometry's primary objective is the study of Euclidean Geometry as a formal, logical system. Where possible, excursions are made into three-dimensional figures and elementary analytic geometry. Some review of algebraic materials may be included. This course begins with developing visualization and some drawing skills. Both algebraic and geometric models are introduced and are further enhanced throughout the course. Proofs are developed slowly in the first half of the course. Various proof formats, including paragraph, flow-chart, and two-column proofs are presented. Students are expected to be actively involved in their own learning. The use of manipulatives is integrated into this course.
Geometry Honors
The Geometry Honors course begins with a strong development of visualization and drawing skills. Both algebraic and geometric models are introduced and are used throughout the course. Proofs are developed slowly in the first half of the year. Various proof formats, including paragraph, flow chart, and two column proofs, are presented. Students are expected to be actively involved in their own learning. Manipulatives, constructions, and the computer program Geometer's Sketchpad are also integrated into this course.
Advanced Mathematics
This course consists of a more thorough treatment of Trigonometry and other selected topics in Algebra 2 with Trigonometry to prepare students for further study in mathematics. Algebra 2 with Trigonometry is a prerequisite. The primary objective of the Advanced Math curriculum is to prepare students for Precalculus. Integral to the learning process is the systematic review of earlier concepts learned in Algebra 2 with Trigonometry and procedures in which students use previously learned skills to develop proficiency with more advanced concepts. The Advanced Math course includes organizational skills, communication, mathematical tools, calculators, hands on activities and group work.
Precalculus
The primary objective of the Precalculus curriculum is to prepare students for Calculus. Integral to the learning process is the systematic review of earlier concepts learned in Algebra 2 and/or Advanced Math and procedures in which students use previously learned skills to develop proficiency with more advanced concepts, especially Trigonometry. The Precalculus course includes exploration, communication, mathematical tools, manipulatives, calculators, hands on activities and group work.
Precalculus Honors
Designed to prepare the more advanced student for Advanced Placement Calculus, this course provides students an honors level study of trigonometry, advanced functions, analytic geometry, and data analysis. A faster pace also allows for the introduction of topics from calculus earlier in the second semester. Limits, continuity, the definition of the derivative, techniques of differentiation, and applications of the derivative are all explored. Applications and modeling are included throughout the course. Appropriate technology is used regularly for instruction and assessment.
Calculus
This advanced course is an introduction to the fundamental topics comprising calculus. Algebraic, trigonometric, and transcendental functions are studied in the context of differentiation and integration. The Calculus curriculum includes exploration, communication, mathematical tools, manipulatives, calculators, hands on activities and group work. At the conclusion of this course, students should be able to use calculus methods in a variety of applications and problem solving situations.
Calculus Honors
This advanced course is an introduction to the fundamental topics of calculus. Algebraic, trigonometric, and transcendental functions are studied in the context of differentiation and integration. The Honors Calculus curriculum is designed to introduce students to the many application of calculus, learn the fundamental rules of calculus, and develop strong problem solving skills. Students will learn how to use technologies such as the graphing calculator, Mathematica, and Excel to investigate various calculus topics and real world problems. Upon completion of this course, students should be well prepared to move onto a first year college level calculus course.
Probability and Statistics
Less rigorous than Pre-Calculus, the primary objective of Probability and Statistics is to offer students an opportunity to continue their mathematical studies in a new area. This course begins with an overview of statistics and includes an investigation of the fundamental laws of probability. It also includes such topics as distributions, sampling, regression, estimation, and hypothesis testing.
Advanced Placement Calculus AB
This is a rigorous Advanced Placement course designed to prepare students for the AP Calculus AB exam in the spring. The course seeks to develop students' understanding of the concepts of calculus, while providing experience with its methods and applications. A multi-representational approach to calculus is employed with concepts, results, and problems being expressed graphically, numerically, analytically, and verbally. The connections between these representations are also explored.
Advanced Placement Calculus BC
This is a rigorous Advanced Placement course that prepares students to take the AP Calculus BC exam in the spring. The course seeks to develop advanced problem solving skills by stressing the application of the concepts covered in the problem solving process. The class requires some vacation assignments that are to reinforce the concepts that have been taught. The class moves quickly and covers all the material outlined by the College Board and is intended for students that have had success in Precalculus or lower levels of Calculus and want to challenge themselves at the highest level.
Advanced Placement Statistics
AP Statistics is the high school equivalent of a one semester, introductory college statistics course. In this rigorous course, students develop strategies for collecting, organizing, analyzing, and drawing conclusions from data. Students design, administer, and tabulate results from surveys and experiments. Probability and simulations aid students in constructing models for chance behavior. Sampling distributions provide the logical structure for confidence intervals and hypothesis tests. Students use a TI-84 graphing calculator, Fathom and Minitab statistical software, and Web-based java applets to investigate statistical concepts. To develop effective statistical communication skills, students are required to prepare frequent written and oral analyses of real data. |
2.Overview of How Maple enhances the
student'slearning in Multivariable
Calculus
3.Illustration and practice with major features
for Multivariable Calculus through example Maple worksheets for the classroom
4.Student Multivariable Calculus Projects in
Maple
Introduction
I
have been teaching our second year (Calculus III and IV) for the last 13 of my
27 years at Saint Joseph's
College.I first became acquainted with
Maple at one of the very early ICTCM conferences.We first adopted Maple V to use in Calculus
and other higher level courses.We are
using Maple 12 this year and will likely upgrade versions next year.The Calculus III and IV classes meet three
times a week.One of those three
meetings is in a computer lab.During
the other two class times, we are in a regular classroom equipped with a
computer with Maple, and projection.
The
four primary areas where we have seen advantages in using Maple for our second
year calculus are:
Computation:For some concepts, such as tangent and normal
vectors, computations can be very cumbersome and thus interfere with conceptual
understanding.Maple can remove the drudgery
of the computations, allowing students to focus on theory, methods, and
applications.
Assignment
Verification (Checking Answers):We still want out students to be able to
carry out computations(show
work!).However,Maple can perform the step by step
calculations as well, giving the students a chance to check their work and find
their errors.
Independent
Exploration and Student Projects:Maple's interface allows students to explore and
write about concepts within the same document.They can easily edittheir Maple
commands as well as their accompanying writing.Maple 2D math encourages student to write their mathematics with correct
mathematical notation.For projects,
the interactive style of the software allows students to start by implementing
a scaled-back portion of a project idea, then iteratively expand upon that
idea.By using Maple's help facilities
with instructor's advice and trouble-shooting, students can develop projects
that are really fun as well as educational.
At our annual Saint
Joseph's College Undergraduate Colloquium, Calculus
IV students have been presenting their projects in an informal walk-through
"poster session".This format gives them
a chance to explain their work to their peers and appreciative faculty in very
creative ways.Traditionally they have
shown their Maple worksheet interactively along with creative posters or a
slide show presentation.They often
have a fun activity for the attendees.Some of the memorable activities include:
Maple
is a real power tool, which means it requires an investment of time to learn to
use its features effectively.Then -- as
soon as one starts to feel comfortable – an upgrade is released with new
features to master.Some of the helpful
resources for keeping up are: |
Book Description: Your guide to a higher score on the Praxis II?: Mathematics Content Knowledge Test (0061)Why CliffsTestPrep Guides?Go with the name you know and trustGet the information you need--fast!Written by test-prep specialistsAbout the contents:Introduction* Overview of the exam* How to use this book* Proven study strategies and test-taking tipsPart I: Subject Review* Focused review of all exam topics: arithmetic and basic algebra, geometry, trigonometry, analytic geometry, functions and their graphs, calculus, probability and statistics, discrete mathematics, linear algebra, computer science, and mathematical reasoning and modeling* Reviews cover basic terminology and principles, relevant laws, formulas, theorems, algorithms, and morePart II: 3 Full-Length Practice Examinations* Like the actual exam, each practice exam includes 50 multiple-choice questions* Complete with answers and explanations for all questionsTest Prep-Essentials from the Experts at CliffsNotes? |
Academics
Jean-Martin Albert - Mathematics Fellow
Jean-Martin Albert is an enthusiastic teacher with experience teaching math courses and tutoring students, and is well prepared to teach mathematical concepts to students coming from a wide variety of backgrounds. He comes to Marlboro after working in several Canadian provinces, and is looking forward to collaborating with logicians in New England and New York.
Teaching Philosophy
From his experience as a math teacher and teaching assistant for all levels of undergraduate courses, Jean-Martin has learned that many students, especially in introductory courses, are intimidated by mathematics, and view the material as abstract and tedious. He says, "I try to show them that mathematics can be fun and useful, and most often, especially at the introductory level, not very far removed from reality. My enthusiasm plays a big part in the process."
Jean-Martin believes that "the best way to learn mathematics is to do mathematics, and a big part of doing mathematics is solving problems. I try to show the students that each formula and each theorem is the answer to a particular question, whether the question came from the real world or is mathematics for its own sake." He also notes that discussion and teamwork are an important part of learning mathematics. "Each student will see a problem in a slightly different light, and so each will consider a different approach to the solution." He then asks students to try to explain to him or to other students how they arrive at a solution, in order to help them internalize and remember the solution they found. He also encourages mistakes: "I want the students to fall into traps, make mistakes, and learn to see exactly where they made the mistake, and how to recover from it."
Scholarly Activities
Jean-Martin is interested in mathematical logic, more specifically continuous model theory and its applications to functional analysis. This is a new and exciting area of mathematical logic, which has applications to many other mathematical areas such as algebra, topology, functional analysis, p-adic analysis and probability.
Selected Conference Presentations
"Strong Conceptual Completeness for First-Order Continuous Logic." North American Annual Meeting of the Association for Symbolic Logic, March 2010. |
Through intensive research and development, Agile Mind Calculus was created for teachers, administrators, and schools seeking to offer this central college-level course to a broad cross-section of students.
Authored by expert educators with extensive AP™ experience, Agile Mind's authoritative resources enable teachers to efficiently manage instructional time, enhance their teaching expertise, and extend their presence with their students outside of the classroom.
Calculus AB follows the well-respected Advanced Placement syllabus and emphasizes algebraic, numerical, and graphical representations throughout. Students will be prepared for success on the AP Calculus exam and in college, with thorough grounding in:
The Calculus Multiple choice and Constructed response have helped my students develop
self-confidence that they will be prepared for the AP exam. It is beneficial to be able to tell them that this particular Multiple choice question or Constructed response task is very similar to these real AP questions, showing them those real test items. |
Do the Math Workbook for Intermediate Algebra
Intermediate Algebra
Intermediate Algebra Plus MyMathLab -- Access Card Package
MathXL Tutorials on CD for Intermediate Algebra
Student Solutions Manual (standalone) for Intermediate Algebra
Videos on DVD for Intermediate Algebra
Summary
Intermediate Algebra is 1-semester gateway course to other college-level mathematics courses. The goal of the Intermediate Algebra course is to provide students with the mathematical skills that are prerequisites for courses such as College Algebra, Elementary Statistics, Liberal-Arts Math and Mathematics for Teachers.
Table of Contents
Preface Chapter R
Real Numbers and Algebraic Expressions
All the Arithmetic Yoursquo;ll Need
Success in Mathematics
What to Do the First Week of the Semester
What to Do Before, During, and After Class
How to Use The Text Effectively
How to Prepare for an Exam
Sets and Classification of Numbers
Use Set Notation
Know the Classification of Numbers
Approximate Decimals by Rounding or Truncating
Plot Points on the Real Number Line
Use Inequalities to Order Real Numbers
Operations on Signed Numbers; Properties of Real Numbers
Compute the Absolute Value of a Real Number
Add and Subtract Signed Numbers
Multiply and Divide Signed Numbers
Perform Operations on Fractions
Know the Associative and Distributive Properties of Real Numbers
Order of Operations
Evaluate Real Numbers with Exponents
Use the Order of Operations to Evaluate Expressions Algebraic Expressions |
Elementary Statistics - 11th edition
Summary: Succeed in statistics with ELEMENTARY STATISTICS! With its down-to-earth writing style and relevant examples, exercises, and applications, this book gives you the tools you need to make the grade in your statistics course. Learning to use MINITAB?, Excel?, and the TI-83/84 graphing calculator is made easy with output and instructions included throughout the text. Need extra help? A wealth of online supplements offers you guided tutorial support, step-by-step video solutions, and imme...show morediate feedback145173.99 +$3.99 s/h
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Description: This course studies fundamental algebraic systems in mathematics, selected from groups, rings, fields, and modules. Examples of groups include the invertible matrices with a fixed size and the roots of unity. Rings are illustrated by integers, polynomials, and modular arithmetic. Complex numbers, rational numbers, and rational functions are examples of fields. (There are also finite fields, which are used all the time in computer science.) Finally, ordinary vectors in space and any lattice in the plane are examples of modules. The concern with these algebraic systems is not simply the study of individual systems, but also of functions between systems which carry one operation into the other. For instance, the determinant not only converts matrices into numbers, but it sends a product of matrices into a product of numbers. The level of attention given to such operation-preserving transformations (putting them on an equal footing with the algebraic systems they transform) is one of the characteristic features of abstract algebra, and also one of the algebraic ideas which have reached into other areas of mathematics. Prerequisites: A grade of C or better in MATH 2142(244) or 2710(213). Recommended preparation: MATH 2210Q(227Q) or 2144Q(246Q). Offered: Fall Credits: 3
These are the most recent data in the math department database for Math 216 in Storrs Campus.
There could be more recent data on our class schedules page, where you can also check for sections at other campuses. |
Elementary Statistics - With Cd - 6th edition
Summary: Elementary Statistics is appropriate for a one-semester introductory statistics course, with an algebra prerequisite. ES has a reputation for being thorough and precise, and for using real data extensively. Students find the book readable and clear, and the math level is right for the diverse population that takes the introductory statistics course. The text thoroughly explains and illustrates concepts through an abundance of worked out examples.
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Core Abilities (Note: since this course may be taken in partial fulfillment of the general education requirements, this syllabus includes the following set of core ability goals.)
1. Thinking: Students engage in the process of inquiry and problem solving that involves both critical and creative thinking.
Students will be exposed to the logic of mathematical proof
Students will develop their problem-solving skills
Calculus is a major intellectual development in human history and students will think through the concepts
2. Communication: Students communicate orally and in writing in an appropriate manner both personally and professionally.
Students will develop their skills of written mathematical communication, specifically learning to properly use the language and notation of the Calculus
Students will develop their verbal mathematical communication skills, both in small groups and in class discussions
3. Life Values: Students analyze, evaluate and respond to ethical issues from informed personal, professional, and social value systems.
Students will see the importance of integrity regarding their own scholarship
4. Community Involvement: Students demonstrate skills of interdependent group participation and decision-making.
Students will work in groups, learning to share their ideas and skills, and respecting the ideas and skills of others
Specific Course Goals:
1. From the perspective of mathematical content, this course should allow the student to expand and apply skills and knowledge gained in the first semester of Calculus to the topics of integration and applications of integration.
2. The student will gain knowledge and skills, and the ability to apply these, to a variety of situations which might be encountered in the world of mathematics, science, or engineering.
3. The student will further improve his/her ability to communicate mathematical ideas and solutions to problems.
4. The student will improve her/his problem-solving ability.
5. From a most general perspective, the student should see growth in his/her mathematical maturity. The three-semester sequence of calculus courses form the foundation of any serious study of mathematics or other mathematically-oriented disciplines.
Course Philosophy and Procedures:
I am a firm believer in the notion that you as the student must be actively engaged in the learning process and that this is best accomplished by your DOING mathematics. My job here is not to somehow convince you that I know how to do the mathematics included in the course, but to help guide YOU through the material; I am to be a "guide on the side, NOT a sage on the stage". I may not always say things in the way which best leads to your full comprehension, but I will try to do so and I ask that you try to see your learning as something YOU DO rather than as something I do to you. PLEASE feel free to come see me to discuss questions or concerns you may have during the course; I will not bite! PLEASE ask questions in class if you have them; the only dumb question is the one you fail to ask!
Homework: Let me therefore urge you to make it a regular part of your day to try working the homework problems. There will never be enough time for us to go through every listed problem in class, and it is probably unrealistic to think that you will be able to find the time to work through every listed problem, but you should at least spend some time thinking about virtually every problem, and working the more interesting or challenging to completion. The daily homework assignments will not generally be collected or graded. They are intended to structure your learning so that you regularly challenge yourself to see that you understand the material we are looking at. The important thing is that you at least look at all the assigned problems. You should also feel free to work other problems if you deem it necessary for your comprehension of the content.
You should view homework assignments as a test to see how well you understand the material and you should bring to the next class any questions you might have.
Group Work: In general, I think students can benefit greatly by working together on problems. While there is some danger of the "blind leading the blind" syndrome, or of students deceiving themselves into thinking they understand the material better than they actually do, for the most part grappling with ideas and trying to explain them to another student or learning to listen to others explain an idea is a wonderful way to see that you really do understand the material. In class we will spend an occasional day working on a group "lab", and we sill also typically have a group "practice exam" before the individual exams, and I also encourage you to find a "learning group" outside of class.
Portfolio: I will be asking you to keep a PORTFOLIO of your work. This portfolio will be collected twice during the semester, once upon our return from our spring break, on Monday 17 March, and again at the end of the course, on Friday 2 May. Each of these portfolios should be a representative collection of your work during that half of the semester; each collection you turn in should include 5 problems, written up in a finished form, along with a brief discussion of why you chose to include that particular problem. These problems should represent work you are proud of, or problems which brought you to a breakthrough point. Each of these portfolios of your work will be worth 40 points, 8 points per problem. Presumably you will hand in 5 sheets of paper, each of which will include a nicely organized solution to the problem, which might be from homework, or from a lab or worksheet, of from an exam, along with that reflection mentioned above.
Grading: I use a rather traditional Grading Scale: A = 90% (or better), AB = 87%, B = 80%, BC = 77%, C = 70%, CD = 67%, D = 60%. I do try to come up with a mix of points so that exams are not weighted too heavily; about two-thirds of the points during the semester will come from individual, in-class exams - the rest will come from group labs, take-home problems, group practice exams, journals, and portfolios.
Late Assignments: It is rarely much of a problem at the level of a calculus course, but it remains important that students turn in work in a timely manner, so that they do not get behind. Consequently, Late Assignments will be penalized 20% of the possible points for each class period late, up to a maximum of three periods. Assignments more than three days late will not be accepted.
Attendance: I do not prefer to quantify your attendance in terms of a grade, but I can assure you that your chances of success will be much improved by regular attendance.
Americans with Disability Act:
If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me or Wayne Wojciechowski in MC 320 (796-3085) within 10 days to discuss your accommodation needs. |
Linear Algebra:Gateway to Mathematics
Description
This text is designed to resolve the conflict between the abstractions of linear algebra and the needs and abilities of the students who may have dealt only briefly with the theoretical aspects of previous mathematics courses. The author recognizes that many students will at first feel uncomfortable, or at least unfamiliar, with the theoretical nature inherent in many of the topics in linear algebra. Numerous discussions of the logical structure of proofs, the need to translate terminology into notation, and suggestions about efficient ways to discover a proof are included. This text combines the many simple and elegant results of elementary linear algebra with some powerful computational techniques to demonstrate that theorectical mathematics need not be difficult, mysterious, or useless. This book is written for the second course in linear algebra (or the first course, if the instructor is receptive to this approach). |
Organized for use in a lecture-and-computer-lab format, this hands-on book presents the finite element method (FEM) as a tool to find approximate solutions of differential equations, making it a useful resource for students from a variety of disciplines.
The book aims for an appropriate balance among the theory, generality, and practical applications of the FEM. Theoretical details are presented in an informal style appealing to intuition rather than mathematical rigor. To make the concepts clear, all computational details are fully explained and numerous examples are included, showing all calculations. All finite element procedures are implemented in interactive Mathematica notebooks, from which all necessary computations are readily apparent. |
examinati... read more
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Our Editors also recommend:The Calculus Primer by William L. Schaaf Comprehensive but concise, this workbook is less rigorous than most calculus texts. Topics include functions, derivatives, differentiation of algebraic functions, partial differentiation, indeterminate forms, definite integral, and much more. 1963A Long Way from Euclid by Constance Reid Lively guide by a prominent historian focuses on the role of Euclid's Elements in subsequent mathematical developments. Elementary algebra and plane geometry are sole prerequisites. 80 drawings. 1963Product Description:
examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.
Reprint of the John Wiley & Sons, Inc., New York, 1967 edition.
A solutions manual to accompany this text is available for free download. Click here to download PDF version now |
Graphing calculator?
Graphing calculator?
I need to get a graphing calculator but I am not sure which one suits me best. I need it for graphing and possibly some more advanced functions. I checked out the prices and the TI-89 is somewhat expensive. It seems a little "hyped" than the other older models. Does the TI-85 have everything the 89 has, just on older hardware?
I've owned a TI-83, a TI-84+, and I currently own a TI-89 Titanium, and I must say that the TI-89 T blows the other ones away.
There is much more memory integrated in the calculator, it is much faster. It is also much easier to use the more advanced functions (IMO), and it looks a lot nicer. I would really suggest a TI-89T, you won't be disappointed.
I'm going with Fragment. I own a TI-89 Titanium and it's a very good calculator, has plenty of features (symbolic differentiation and integration, simple ODEs, 2D curve sketching, 3D graphing, simultaneous linear equation solver...), and the learning curve isn't steep at all.
A word of warning, though - don't get addicted to it. If you're still in high school of your first year of college, try to work out stuff by hand. If you keep using your calculator to do simple calculations, you're gonna go rusty fast.
What kind of class do you need this calculator for? For other stuff, I use the European equivalent of this calculator, and it does plenty of stuff (but it doesn't have graphing capabilities), like numerical integration and differentiation, quadratic and cubic equations, up to 3 simultaneous equations, complex numbers, statistical functions, ... |
Measure and Integration for Use
9780198536086
ISBN:
0198536089
Pub Date: 1985 Publisher: Oxford University Press, Incorporated
Summary: Although of unquestioned power and practical utility, the Lebesgue Theory of measure and integration tends to be avoided by mathematicians, due to the difficulty of obtaining detailed proofs of a few crucial theorems. In this concise and easy-to-read introduction, the author demonstrates that the day-to-day skills gleaned from Legesgue Theory far outweigh the effort needed to master it. This compact account develops ...the theory as it applies to abstract spaces, describes its importance to differential and integral calculus, and shows how the theory can be applied to geometry, harmonic analysis, and probability. Postgraduates in mathematics and science who need integration and measure theory as a working tool, as well as statisticians and other scientists, will find this practical work invaluable.[read more] |
geometr... read more
Regular Polytopes by H. S. M. Coxeter Foremost book available on polytopes, incorporating ancient Greek and most modern work. Discusses polygons, polyhedrons, and multi-dimensional polytopes. Definitions of symbols. Includes 8 tables plus many diagrams and examples. 1963 edition.
Shape Theory: Categorical Methods of Approximation by J. M. Cordier, T. Porter This in-depth treatment uses shape theory as a "case study" to illustrate situations common to many areas of mathematics, including the use of archetypal models as a basis for systems of approximations. 1989 editionProjective Geometry by T. Ewan Faulkner Highlighted by numerous examples, this book explores methods of the projective geometry of the plane. Examines the conic, the general equation of the 2nd degree, and the relationship between Euclidean and projective geometry. 1960 edition.
Non-Riemannian Geometry by Luther Pfahler Eisenhart This concise text by a prominent mathematician deals chiefly with manifolds dominated by the geometry of paths. Topics include asymmetric and symmetric connections, the projective geometry of paths, and the geometry of sub-spaces. 1927 edition.
The Elements of Non-Euclidean Geometry by D. M.Y. Sommerville Renowned for its lucid yet meticulous exposition, this classic allows students to follow the development of non-Euclidean geometry from a fundamental analysis of the concept of parallelism to more advanced topics. 1914 edition. Includes 133 figuresProduct Description:
geometry plays in a wide range of mathematical applications.
Bonus Editorial Feature:
Harold In the Author's Own Words: "I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."
"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."
"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."
"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter Harold |
Tobin A. Driscoll
Other Titles in Applied Mathematics 115
This engaging book is a concise introduction to the essentials of the MATLAB programming language.
Learning MATLAB is ideal for readers seeking
• a focused and brief approach to the software; • numerous examples and exercises involving the software's most useful and sophisticated features; and • an overview of the most common scientific computing tasks for which the software can be used..
The presentation is designed to guide new users through the basics of interacting with and programming in the MATLAB software, while also presenting some of its more important and advanced techniques, including how to solve common problem types in scientific computing. Rather than including exhaustive technical material, the author teaches through readily understood examples and numerous exercises that range from straightforward to very challenging. Readers are encouraged to learn by doing: entering the examples themselves, reading the online help, and trying the exercises.
Audience This handbook is suitable for graduate students, advanced undergraduate students, and professional researchers in mathematics, scientific computing, and application areas in science and engineering. It can be used as the primary text for a short course, as a companion textbook for a numerical computing course, or for self-study.
About the Author Tobin A. Driscoll is an Associate Professor in the Department of Mathematical Sciences at the University of Delaware whose research focuses on the numerical analysis of differential equations. He is coauthor with L. N.Trefethen of Schwarz–Christoffel Mapping (Cambridge University Press, 2002). |
...CalcSharp is .Net application that evaluates mathematical expressions step by step.
It uses a different approach from a conventional calculator, which is more
natural to the way people calculate. When...
...Panageos: This is a powerful Plane Analytic Geometry Problem Solver and Visualizer.
Panageos is oriented to the intensive solution of problems on Plane Analytic Geometry
The main feature of Panageos...
...The program allows you calculate pension annuities and/or pension contributions for non-state pension funds, based on actuary mathematics. The software can use various gender stats, investment rates to...
...Graph Magics - an ultimate tool for Graph Theory. Containing a numerous collection of functions, utilities and algorithms, it offers you the possibility of easy, fast and efficient construction, modification and...
...SplineCalc is multipurpose scientific calculator for implementation of various mathematical operations with maps (grids), data tables, lines, polygons and numbers. You can use arithmetical and logical operations with any above...
...Smooth Operators is a complete solution for learning, practicing, and testing the order of operations. An interactive lesson teaches the math concepts thoroughly with explanations and example problems, including an...
...Highly interactive tutorials and self-test system for individual e-learning, home schooling, college and high school computer learning centers, and distance learning. The product covers the standard topics of... |
Precalculus College Algebra, Algebra and Trigonometry, Trigonometry, and Precalculus. A proven motivator for students of diverse mathematical backgrounds, this text explores mathematics within the context of real-life, The... MORE Trigonometric Functions; Analytic Trigonometry; Analytic Geometry; Counting and Probability; A Preview of Calculus; and more. For individuals with an interest in learning Precalculus as it applies to their everyday lives. |
Students, your effort is the single most important factor in determining future success. Math is a skill that is developed through practice. While I can expose you to new ideas and help you perfect your problem solving skills, you must put in practice time. In this course, WebAssign plays a vital role in providing practice with immediate feedback at any hour of the day or night. Forming a study group and working together to discuss and complete assignments is an excellent strategy for success.
You should expect to spend between two and three hours a week on homework assignments. If the assignments are taking longer than that, you need to attend extra help!
You should be able to complete the assignments with at least 80% accuracy. If you are struggling with the assignments, you should attend extra help ASAP. In extra help you will be given an extension, additional tries and help. Please bring your written work with you so we can quickly pinpoint the "difficulty".
If you are thinking, "I'm good at math and I've never had to do homework before" or "Homework is busy work and a waste of time." you are in for a rude awakening.
Parents, look at homework grades on Powerschool. They will let you know if your child is making the required effort. Since homework averages are almost always better than test and quiz averages, homework almost always helps a student's overall average. |
Product Synopsis
Written by well-respected authors, the Cambridge Checkpoint Mathematics suite provides a comprehensive structured resource which covers the full Cambridge Secondary 1 Mathematics framework in three stages. This Teacher's Resource for Stage 7 offers advice on how to introduce concepts in the class, and gives ideas for activities to help engage students with the subject matter. Answers to all questions in the Coursebook and Practice Book are also included |
Product Description
This Saxon Algebra 2 Home Study Kit includes the Student Textbook, Testing Book and Answer Key. Traditional second-year algebra topics, as well as a full semester of informal geometry, are included with both real-world, abstract and interdisciplinary applications. Topics include geometric functions like angles, perimeters, and proportional segments; negative exponents; quadratic equations; metric conversions; logarithms; and advanced factoring. Student Text is 558 pages, short answers for problem/practice sets, an index and glossary are included; hardcover.
The Test book contains both student tests and solutions with work shown along with the final answer. 32 tests are included.
The Answer key shows only the final solution for the practice and problem sets found in the student text. 44 pages, paperback.
Product Reviews
Is a great product.
Is a very good learning book. The book my child enjoys.
September 10, 2012
I like the way the children can go back and find the example if they have forgotten how to solve the problem.It is agood set to teach by.
June 14, 2012
The books coupled with the CD-Rom help my son.
I recommend obtaining the CD-Rom or other CD teaching kits with the books to gain a more well and full rounded math education. If your child enjoys being somewhat independent the combination works well.
November 3, 2011
I am not a full time home-schooling Mom but I have found that when the kids work on their next year's math during the summer months, it REALLY aides the transition into the more intense demands of pre-college math.
July 16, 2011 |
Objectives
Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.
This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. If you would like a quick review of arithmetic before attempting the study of algebra, this chapter is recommended reading. If you feel your arithmetic skills are pretty good, then move on to Basic Properties of Real Numbers ((Reference)). However you feel, do not hesitate to use this chapter as a quick reference of arithmetic techniques.
The other chapters include Practice Sets paired with Sample Sets with sufficient space for the student to work out the problems. In addition, these chapters include a Summary of Key Concepts, Exercise Supplements, and Proficiency Exams |
This year long course provides students with the applications and concepts necessary for success in pre-algebra. Concepts addressed include number patterns and algebra, operations with decimals, percents and fractions, data analysis and probability, integers, geometry and measurement.
This year long course provides students with the applications and concepts necessary for success in pre-algebra. Concepts addressed include algebraic concepts, operations with decimals, percents and fractions, real world applications, data analysis, algebraic concepts, geometry and measurement.
This year long course provides students with pre-algebra skill development necessary for success in Algebra 1. Concepts addressed include algebraic concepts such as expressions, equations, functions, real world applications, and data analysis.
Textbook(s): Glencoe or Algebraic Thinking Pt. 2
Algebra 1:
Prerequisites:Pre Algebra and Orleans Hanna test
Algebra 1 provides students with the materials outlined in the Maryland Core Learning Goals in Algebra 1 and Data Analysis. These goals include indicators that require experiences with problem solving and patterns, graphing linear equations, finding quadratics and other non-linear functions. Students will take the Algebra/Data Analysis High School Assessment at the end of this course as a part of the high school graduation requirement. The middle school Algebra 1 student will also take their grade level MSA assessment.
Textbook(s): Algebra 1, Prentice Hall
Geometry:
Prerequisite:Successful completion of Algebra 1, Grade 7
Geometry provides students with the skills outlined in the Maryland Core Learning Goals for Geometry. These skills include using logic to develop arguments, working with postulates and theorems of Euclidian geometry, applying rules for parallel and perpendicular lines, identifying congruent and similar figures, classifying polygons, measuring angles and writing proofs of triangle congruence, drawing, constructing and performing plane transformations.
Textbook(s): Geometry, Prentice Hall
Last modified: 9/5/2007 10:01:07 PM
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Overview The workbook provides teachers and students with 143 pages of challenging worksheets for practice to help develop Geometry skills. The workbook includes all topics covered throughout a Geometry course. The worksheets contained in this workbook consist of several different formats to provide a variety of exercises. Formats include multiple choice questions, fill-in the blank, problem solving, puzzle worksheets, and more.
Geometry Workbook Teacher Edition 1.0 - Loose Leaf Bound This version is loose leaf bound for easy removal/replacement of worksheets from the workbook. This binding method allows for flexible use of the worksheets including copying as needed under license by the author as specified in the copyright notice. This edition of the workbook contains answers to all of the problems as a key in the back of the workbook.
Geometry Workbook Teacher Edition 1.0 - Spiral Bound This version is spiral bound to prevent loss of worksheets from the workbook. This binding method allows for easy access to the worksheets with protection against loss of individual pages. This edition of the workbook contains answers to all of the problems as a key in the back of the workbook.
Geometry Workbook Student Editions 1.0 This version can be ordered either in the loose leaf bound version or in spiral bound version. This edition contains all workbook pages, however there is not an answer key located in the back of the workbook. |
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Are your calculus pupils aware that they are standing on the shoulders of giants? This lesson provides a big picture view of the connection between differential and integral calculus and throws in a bit of history, as well. Note: The calculus controversy paper is not included but one can find a number of good resources on the Internet regarding the development of calculus and the role of Newton and Leibnez.
This video covers the differential notation dy/dx and generalizes the rule for finding the derivative of any polynomial. It also extends the notion of the derivatives covered in the Khan Academy videos, �Calculus Derivatives 2� and �Calculus Derivatives 2.5 (HD).� Note: Additional practice using the power rule for differentiating polynomials (including some with negative exponents) is available to the listener.
This video is the 4th in a series of videos that explains derivatives. First, Sal shows another example of differentiating a polynomial and then shows two examples using the chain rule. Sal continues the chain rule in the next video.
This video continues with examples of differentiating functions using the chain rule including examples that use negative exponents and more complicated nested parenthesis. Note: Additional practice on using the chain rule is available.
Continuing to use the chain rule, Sal shows more examples of finding the derivative, this time, by looking at composite functions. Note: The current set of practice problems titled �Chain rule 1� cannot be solved until one knows how to find the derivative of ex and trigonometric functions
In the first example, instead of actually using the quotient rule, Sal rewrites the denominator as a negative exponent and uses the product rule. In subsequent examples, Sal shows, but does not prove, the derivative of several interesting functions including ex, ln x, sin x, cos x, and tan x.
After defining L�Hopital�s rule, Sal shows an example of using the rule to solve for the limit as x approaches 0 of (sin x) / x. He also describes what it means for a fraction to have indeterminate form.
After applying L�Hopitals Rule once and showing that the limit is still indeterminate, Sal applies the rule a second and third time before arriving at a limit that exists. He then shows that given that the final limit exits, so do the previous ones including the limit of the first original problem. Note: Practice problems on L�Hopital�s rule are available and can be practiced now or after watching the other example videos. |
Math 123: Foundations of Elementary Geometry
Course Description
Math 123 is intended for prospective and current elementary school teachers. The course examines problem-solving techniques and mathematics related to topics taught at the K–8 level. Topics include measurement, geometry and the use of technology.
Who should take this course?
Prospective and current elementary school teachers. You should consult the planning sheet for your program and consult an advisor to determine if this course is appropriate for you.
Who is eligible to take this course?
The prerequisite for this course is Math 121 with a grade of 2.0 or higher.
Is this course transferable?
This course may transfer to certain universities if the student enrolls in a teacher-preparation program; consult an advisor or see the Transfer Center to determine transferability.
What textbook is used for this course?
The tenth edition of A Problem Solving Approach to Mathematics for Elementary School Teachers by Rick Billstein, Shlomo Libeskind and Johnny W. Lott. |
Balance Benders Logic and Algebraic Reasoning Puzzles: Level 1
By Robert Femiano
Level 1: Grades 4-12+
This book can be used as quick, fun logic problems or as stepping-stones to success in algebra. Students develop deductive thinking and pre-algebra skills as they solve balance puzzles that are more fun and addictive than Sudoku puzzles. Each balance must be analyzed to identify the clues, and information synthesized to solve the puzzle. 48 pgs each. |
Determinant
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.
Matrices
In mathematics, a matrix is a rectangular table of elements, which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters.
Matrices are described by the field of matrix theory. Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra.
The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (written m × n) and m and n are called its dimensions. |
Hello math gurus. This is my first post in this forum. I struggle a lot with free math patterns and functions worksheets for second grade problems. No matter how hard I try, I just am not able to solve any question in less than an hour. If things go this way, I fear I will not be able to pass my math exam.
Don't fret my friend. It's just a matter of time before you'll have no trouble in solving those problems in free math patterns and functions worksheets for second grade. I have the exact solution for your math problems, it's called Algebra Buster. It's quite new but I assure you that it would be perfect in assisting you in your algebra problems. It's a piece of software where you can answer any kind of algebra problems with ease. It's also user friendly and displays a lot of useful data that makes you learn the subject matter fully.
I discovered a some software programs that are pertinent. I verified them. The Algebra Buster appeared to be the best suited one for adding fractions, inverse matrices and radical inequalities. It was also effortless to work with. It took me step by step towards the solution rather than only giving the answer. That way I got to learn how to explain the problems too. By the time I was finished, I had learnt how to explain the problems. I found them of use in Basic Math, Basic Math and College Algebra which helped me in my algebra classes. May be, this is just what you want. Why not try this out?
A truly piece of algebra software is Algebra Buster. Even I faced similar problems while solving radical expressions, converting fractions and powers. Just by typing in the problem workbookand clicking on Solve – and step by step solution to my algebra homework would be ready. I have used it through several algebra classes - Basic Math, Basic Math and Algebra 1. I highly recommend the program. |
Getting Started with "Math in ONE"
Code use for Windows, Android versions
If no symbol is indicated, then instructions apply to both Windows and Android products.
(W)
There are two main resources to help you learn about "Math in ONE" which are found in the "Reference Manual" and the convenient descriptions built into the program. Two examples are presented below.
When using "Math in ONE", these descriptions appear with the right click of your mouse on an object (e.g. checkbox, button,...), then left click on the pop up button. Refer to the example on the left.
For all the pull down menus (e.g."Functions"), simply place your mouse over an item to review its description. Refer to the example on the right.
Thank you for your interest in "Math in ONE"! Please refer to this reference manual for further guidelines. We appreciate all of your feedback regarding this site and the "Math in ONE" program. We will make all the effort to satisfy you. We hope you enjoy working with this product! |
Category: Math curriculum in general
Some people will be surprised that our Mathematical Literacy course includes some factoring. Over the years, the topic of factoring has been a focal point of conversations — almost with the assumption that a reform math course would not cover any factoring. Sometimes, we go to the extreme view of "anything not practical right now … will be omitted", and factoring is usually not very practical.
In our Mathematical Literacy course we covered factoring last week — true, this is just the GCF (no trinomial methods nor special formulas). Since we only include GCF as a method students have an easier time. However, if we had time, I would not mind if we covered a little more factoring.
For language skills, it is important that people be able to express thoughts concisely (simplify); in some important situations, it is even more important to be able to express thoughts in a more complex way that maintains the equivalent message — persuasive writing and speaking are particular modes in this style. In a general way, learning (or a process) that can only be used one direction is usually learned only partially. Deeper learning depends upon a variety of experiences with objects or ideas.
Factoring plays a comparable role in any course emphasizing algebraic reasoning. A basic issue in algebraic reasoning is "Adding or multiplying?" Many of our students believe that parentheses always show two things — what to do first (under the curse of PEMDAS) and "this is a product". Our work with the GCF puts students right in the middle of this confusion; in other words, the GCF is a great opportunity for students to better understand basic algebraic notation.
Of course, one risk of this work with the GCF is that students get even more confused. We need to be careful that assessments help students understand better; within the Math Lit class, I need more experience designing the class work so better assessments can be delivered to students.
Of the traditional developmental algebra content, factoring is not my lowest priority — it connects with basic issues of algebra. I can't say the same thing for radical expressions, where we deal with procedures only vaguely connected with exponents. I also place 'rational expressions' lower in priority than factoring; outside of the very basic ideas of reducing simple rational expressions, our time on operations and equations with rational expressions list mostly wasted … the emphasis ends up on procedures, not concepts and understanding. Such topics have been included in developmental courses because they are seen as needed in pre-calculus courses … because they are seen as needed in calculus courses. We should strengthen this flimsy curriculum design based on student needs AND content needs in deliberate ways.
All of us have a role in this process so that mathematics becomes an enabling process rather than a inhibiting process. Factoring polynomials is not necessarily an evil to be avoided.
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The AMATYC conference (Anaheim, October 31 to November 3) will include several sessions on New Life-inspired efforts to reform developmental mathematics. I will be doing a general session on the New Life model at the conference, and other sessions will focus on particular implementations. Over the next month or so, I will be posting a detailed schedule.
This conference will not include a workshop on the New Life courses; this workshop was done at last year's conference and the materials are still available at If you want to know more about the details of MLCS (Mathematical Literacy for College Students) and AL (Algebraic Literacy), I plan to create some additional 5-minute presentations about each — they will be posted on the "Instant Presentations" page (
Some related work will also be available at the conference — the Dana Center "New Mathways Project" will have sessions. When the mini-program is available, I will post a summary of all reform-related sessions.
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A course design can facilitate learning, and a course design can hinder learning. I suspect that we get so focused on the details of our math courses that we may not notice whether our course is facilitating or hindering.
In our Math Literacy class, we have been working on algebraic reasoning. On the surface, the class looks like we are not 'covering' very much because we don't include some typical algebraic (developmental) topics. We found some hidden treasure this week in class.
As we often do, part of class is based on groups figuring out problems with some guidance and reflection. Today this meant that we had each group do an equation 'tag-team' style — each student could either do the next step, or erase the last step. Students had a little trouble playing by the rules, and wanted to switch to 'their' method to solve the equation. The payoff came when we talked about the different choices, as more students figured out that they have options for linear equations.
The hidden treasure came next, not that students saw it as totally good. We looked at how we could solve equations of a type never seen before, starting with a simple rational equation (namely, 5 = 200/x). Students could see the solution (40) though not always obtained formally, so we talked about doing 'opposite' operations to solve. We followed this with a radical equation (the pendulum model), which is not normally seen in this level of math course. To solve for the length inside the radical, we listed the calculation steps if we knew the length and wanted to calculate the period. Then, we reversed — the opposite operations in the reverse order.
To me, the hidden treasure in this is that students get to think about both types of skills that we use in mathematics — we have routine procedures (often based on properties) and we have reasoning about statements (often based on relationships unique to the problem). Wouldn't it be wonderful if students developed both strategies, instead of just using routine procedures (often memorized)?
It's clear that my hidden treasure was not perfectly clear to students; after this discussion, we had a worksheet which included an equation of related design. They generally understood the reverse order idea, but thought they should do them in a different order — a choice which requires applying properties of expressions. Our conversation was more satisfying than normal because we had used the reasoning approach, and talked about choices.
Students may still 'want' a recipe for solving equations and simplifying expressions. Giving students a recipe hides the math treasure; emphasizing choices and reasoning allows for the possibility of students finding our hidden treasure. |
A course designed to develop the skills and understanding contained in the second year of secondary school algebra. Topics include review of properties of real numbers, functions, algebra of functions, inequalities, polynomials and factoring, rational expressions and equations, radical expressions and equations, quadratic functions and their graphs, solving quadratic equations, and exponential functions. The same course is offered in a two hour (0290) format.
This course is not for college-level credit.
Prerequisites: C or better in MATD 0370 or its equivalent knowledge, or appropriate score on the ACC Mathematics Assessment Test taken before enrolling in ACC mathematics courses. Course Type: D |
Proofs in Algebra: Properties of Equality
In this lesson our instructor talks about proofs in algebra and properties of equality. She talks about addition property of equality, subtraction property of equality, multiplication property of equality, division property of equality, and addition property of equality using angles. She the talks more about the reflexive property of equality, symmetric property of equality, transitive property of equality, substitution property of equality, and the distributive property equality. She discusses the two column proof and does three proof examples. Four complete extra example videos round up this lesson.
This content requires Javascript to be available and enabled in your browser.
One way to organize deductive reasoning is by using a two-column proof
Proofs in Algebra: Properties of Equality
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. |
Understandable Statistics (Hardcover)
9780618949922
ISBN:
0618949925
Edition: 9 Publisher: Houghton Mifflin Company
Summary: This algebra based text is a thorough yet approachable statistics guide for students. The new edition addresses the growing importance of developing students' critical thinking and statistical literacy skills with the introduction of new features and exercises |
97818556444 Understand and Use Mathematics for Derivatives: Advanced Modelling Methods (v. 2)
These two companion volumes offer a comprehensive guide to the 'new maths', analysing and explaining the behaviour of the markets and providing a practical guide to the key mathematical models underlying trading and risk management. Volume II - Advanced Modelling Methods - offers a comprehensive explanation of new terms and techniques used in financial analysis. Ten chapters evaluate the latest developments in the field, formulae calculation and analysis of the different ways in which data can be interpreted and profited from including: computational finance; trading techniques; financial engineering; non-linear maths; mathematical analysis and risk management; statistical inference, skewness and kyrosis; high frequency data and why banks use it; fractals and chaos theory in financial markets |
When you have the right math teacher, learning math can be painless and even fun! Let Basic Math and Pre-Algebra Workbook For Dummies teach you how to overcome your fear of math and approach the subject correctly and directly. A lot of the topics that probably inspired fear before will seem |
Note: This course serves as a pre-requisite for MATH 110 (College Algebra), MATH 130 (Introductory Statistics), or MATH 155 (Mathematics, A Way of Thinking). You must earn at least a "C" grade to qualify for the next course in your sequence.
ALEKS Student Access Code: Purchased from the Viterbo Bookstore.
You will also need the appropriate Course Code for your specific section, which is provided later in this document.
This textbook is published by McGraw-Hill, who also handles ALEKS for institutions of higher education. Our text has been precisely integrated with ALEKS, so that you can use your book for explanations, worked examples and practice problems as we move our way through the course material.
Course Goals and Student Outcomes:
1. Students will demonstrate their readiness for learning algebra.
(a) Students will take ALEKS assessment.
(b) Students will work through pre-algebra ALEKS modules indicated as necessary.
2. Students will improve their mastery of algebraic skills.
(a) Students will take ALEKS assessment of algebra knowledge and skills.
(b) Students will work through the ALEKS modules indicated as necessary.
(c) Students will take indicated exams to demonstrate their learning.
3. Students will develop their ability to apply algebraic thinking and procedures to problem solving.
(a) Students will work through the ALEKS modules that focus on problem solving.
Course Procedures and Policies:
MATH 001: Math 001, "Introductory Algebra", is a not-for-graduation-credit course intended to prepare students for the various courses for which 001 is a pre-requisite, namely MATH 110 (College Algebra), MATH 130 (Introductory Statistics), and MATH 155 (Mathematics, A Way of Thinking). The material is essentially the first year of algebra, which would typically be taken in high school, which explains why this course is numbered 001, and why the 4 credits you will earn here do not count toward graduation, even though they do count toward full-time status.
Your placement score indicated that you have not mastered this content, whatever the reason. To make the best of the situation, your goal here must be to learn this material and master the necessary skills so that you can be successful in the courses you eventually need to take as part of your college program.
ALEKS: ALEKS (Assessment and LEarning in Knowledge Spaces) is a web-based program designed to carefully assess what students know and what they are ready to learn, and then to methodically tutor them in the given material, in this case Introductory Algebra. After registering, you will begin by going through a brief tutorial on the use of the ALEKS input tool, also called the "Answer Editor." On the second day you will do the Initial Assessment in class.
Probably the best thing about ALEKS is that it allows each student to take a course specifically designed for his/her individual needs – students will be working at their own pace and working on material they are ready to learn. The implication of this is that I will not be "lecturing" on textbook sections in the customary way. My role as instructor here is to monitor your learning and to engage in individual tutoring as the need arises.
Another advantage to using ALEKS is that since it is web-based you can work on your course anywhere you have internet access. ALEKS will remember where you left off and will always make sure that you have shown readiness before presenting new material. However, the Initial Assessment and all Quizzes/Exams must be taken during class.
Be sure to do your own work! Your best preparation for online in-class quizzes is when you have been working with ALEKS yourself. By allowing someone else to do your work for you, the only person you are cheating is yourself.
Grading System: Your grade will be determined by the following five factors:
(5) Attendance/ALEKS time outside of class: Important determination for borderline grades
This makes for a total of 560 points. Grades will be assigned according to the scale:
A 93% or above
AB 88 – 92%
B 82 – 87%
BC 77 – 81%
C 70 – 76%
CD 65 – 69%
D 60 – 64%
F < 60%
**NOTE** You also need to complete at least or 80% of the ALEKS topics to pass the course. You need at least a "C" grade to be allowed to advance to the next course in your sequence.
(1) Check-Offs: You will all receive a packet containing Check-Offs which are basically practice problems for questions covered throughout the course. Think of these as homework problems that need to be turned-in. As you master the topics covered on each Check-Off, complete the questions and submit your packet for review. Each section (A – L) is worth 5 points. The Check-Offs need to be completed during class in order to receive credit.
(2) ALEKS Quizzes: In addition to automatic assessments produced by ALEKS based on your completion, ALEKS has the ability to construct exams at points indicated by the instructor. I tell ALEKS what material I want covered and the program constructs problems that test understanding of that material. You will need to complete a Quiz for each chapter of ALEKS, which should correspond to a chapter in your text. Each quiz will be worth 20 points. You can take the quizzes when you complete a chapter, however, there will be a due date at which point you must have the quiz completed whether or not you have finished the material on ALEKS. You can use your handwritten notes on the quizzes. DO NOT use the text.
Think of quizzes as a way to keep yourself on schedule for completing the course by the end of the semester.
* For each Quiz, although it will be taken online, you need to turn-in a paper/pencil copy of the questions with your work and answers at the end of the class period during which each Quiz is taken. At the completion of the Quiz, you will receive your score. If you do not pass a Quiz or plan to retake it, do not turn in your work. Study from it - - you are allowed one chance to re-take each quiz.
**REMINDER**These quizzes, even the re-takes, must be taken in the classroom!
A calculator is allowed for all in-class assessments EXCEPT for Quiz #1 .
_______________ __Quiz Schedule: _____________________
Due Date
Review & Chapter 1 January 24, 2007
Chapter 2 February 7
Chapter 3 February 14
Chapter 4 February 21
Chapter 5 March 14
Chapter 6 March 21
Chapter 7 March 28
Chapter 8 April 11
Chapter 9 April 18
Chapter 10 May 1
(3) ALEKS Modules Completed: On the last two days of class you will take a final assessment, triggered by me. The percentage you score on that assessment is the number of points you receive, based out of 100. This assessment must be completed in the classroom. If you do not finish in one class period, you MUST NOT log on to ALEKS again until you return the next day for class, at which time you can complete your assessment. You must leave record with me as to which problem you were on when you leave, and if you are not on that same problem when you return, you must take the assessment again in the Learning Center. You can use your handwritten notes. DO NOT use the text.
(4) Midterm and Final: The Mid-Term and Final will be a traditional hard-copy test. It will not be an online assessment. Some of you may not be on schedule for the assessments, and this will no doubt affect your performance and, in turn, your Mid-Term and final course grade. The lesson learned here is that part of success in a course is learning the material within a designated amount of time. You will be allowed one instructor-provided note card for notes to use with each test.
(5) Attendance: A major factor in learning mathematics is a regular and focused schedule of practice. You need to practice virtually every day, and for a considerable amount of time each day in order to establish a solid foundation in algebra. To help you work on ALEKS, classroom attendance is REQUIRED every day. I will be keeping track, and will contact you if you miss too many classes. Your attendance will become a factor if your grade falls on a border between two grading categories.
ALEKS Time: ALEKS keeps track of how much time you have put in as well as how much progress you have made. It is almost impossible to pass the class if you do not spend additional time outside of class on ALEKS. It is suggested that you spend an additional 2 to3 hours outside of class for each hour in class. This commitment will help ensure successful progress through ALEKS. I will be keeping track of your hours spent on ALEKS, and I will contact you if it appears you are not putting in enough time outside of class. Your total number of hours on ALEKS will become a factor if your grade falls on a border between two grading categories.
Schedule: Your starting point and rate of progress are based on your initial assessment and learning rate. Because ALEKS allows students to work at their individual pace, students will be at a variety of places in the material throughout the semester. Still, in order to pass the course and move into the subsequent course you will need to demonstrate sufficient knowledge of the material within the semester's time constraints.
It is possible that some of you will actually complete the ALEKS course before the calendar indicates the semester is over, and that's fine. I will still have you take the final exam with the rest of the class on the scheduled date. And it is possible that some of you may reach December without completing the material. ALEKS offers a guarantee that if you do not pass the course despite having put in at least 80 hours, your license to use ALEKS can be renewed for a semester at no cost. In this case, you will be given a grade of "I" (Incomplete), allowing you to work towards completion of the course during the next semester. Of course, this is far from ideal since it means you could not yet enroll in the course you need to take for your major. Use the Quiz dates as a goal for completion!
Americans with Disabilities Act: If you are a person with a disability and require any auxiliary aids, services, or other accommodations for this class, please see me and/or Wayne Wojciechowski, the campus ADA coordinator (MC 335, 796-3085), within ten days to discuss your needs.
Academic Honesty: Per University policy (handbook page 131), you are expected to do your own work for this class. This includes and is not limited to the completion of all ALEKS work, including practice within ALEKS, and assessments. One example of dishonest behavior would be allowing another student to work problems for you in ALEKS. A second example would be having another student take all or part of an online assessment for you. If it is suspected that you violated this policy, you will need to retake the assessment under supervision.
Important Dates:
Tuesday, May 1: Check-Offs Due
Wednesday, May 2 & Thursday, May 3: Completion Assessment on ALEKS
Friday, May 11: Course Final
No Class Meetings:
Monday, March 5 – Thursday, March 8: Spring Break
Monday, April 9: Easter Break
The policies and outline of this course are subject to change at the discretion of the instructor.
Revised 1/9/07
Using ALEKS
You will need the ALEKS Student Access Code on the back of your ALEKS Users' Guide, purchased from the Bookstore. You will also need the Course Code for your section, which is listed below.
1st Type the URL line of your browser. Then add the site to Your Favorites.
ALEKS will lead you through the process of creating an account. Above the Registered Users box you should click:
New User? Sign Up Now!
2nd To enroll in your specific section, you will need the appropriate course code:
WVT63-HTFFX Section 001 MRC 316
On the first day of class, each of you will log in and we will examine the basics of using ALEKS. I will ask you to work your way through the "Answer Editor" tutorial so that you become familiar with how to enter mathematical expressions for assessments, on-line work and quizzes. Then on the second day of class I will have you take the initial ALEKS assessment to get a baseline rating of your skills and readiness for the material in this course. It is important that you always put forth your best effort when taking assessments, because this is how ALEKS determines whether or not you have mastered the material already learned.
ALEKS keeps track of (and lets your instructor see) how much you have mastered and what you are ready to learn. Below are the topics covered in this course. |
Are you afraid of Mathematics? Are you looking for a quality online learning for maths? Do you want homework help, concepts & assignments, CBSE exam preparation, solved exercises from NCERT books? LearnNext is the best solution. Subscribe to LearnNext CBSE software for excellent online learning and tutoring experience.
As students, we always have serious trouble with mathematics. Though India is renowned for its mathematical genius, students often complain about the difficulty with the Maths curriculum. This complaint is more prevalent among CBSE students. LearnNext is developed by teachers who have a thorough grasp of student mind. Our LearnNext is oriented to give thorough preparation and practice of all theorems, formulae of CBSE syllabus through CBSE sample papers and previous papers to be able to tackle Maths. Students can know how many marks are allotted for each chapter like Number Systems, CBSE Algebra, Trigonometry, Coordinate Geometry, Geometry, Mensuration, CBSE Statistics and Probability etc.
Marks distribution of mathematics question paper based on level of difficulty with 12 marks for Easy, 56 and 12 for Average and Difficult questions respectively. At LearnNext, students can master the concepts of maths with the help of CBSE board papers, latest syllabus details, new CBSE marks patterns, question papers, links to latest NCERT and CBSE books. Sound knowledge of maths helps in higher scores for not only CBSE exams but also exams like IIT, AIEEE and other competitive exams. We have a team of expert Maths teachers, who can help students with CBSE Maths problems, questions, doubts, solutions etc for Class X and Class IX maths CBSE. Logon to LearnNext and excel in Maths. |
1567978 / ISBN-13: 9780321567970
Mathematics All Around
"Tom Pirnot" believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking ...Show synopsis"Tom Pirnot" believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking behind the subject matter, so that students are able to truly understand the material and apply it to their lives. This textbook maintains a conversational tone throughout and focuses on motivating students and the mathematics through current applications. Ultimately, students who use this book will become more educated consumers of the vast amount of technical and mathematical information that they encounter daily, transforming them into mathematically aware citizens.Hide synopsis
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Description:Very good. Annotated Edition. Ships SAME or NEXT business day....Very good67970 Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9780321567970 Mathematics All Around
I cannot believe that all of this was crammed into ONE 8 week college course. I'm still getting over the stress of trying to make it through this course. The book is ok if you understand math, but if you don't, you're just going to be more lost than you were before you started. Very confusing stuff |
Here is the information you need in order to be prepared for class and have a successful year.
Supplies: Starting with the first day of school, you will need pencils, a calculator (TI-84 graphing calculator required for Calculus and suggested for Advanced Math), and a 3-ring binder (with paper and 4 dividers)
Notebooks: You are responsible for keeping aNotebook as follows:
·Section 1 - Class notes: organized by date.
·Section 2- Homework: name, assignment (pg. or worksheet #), date.
·Section 3– Quizzes/tests
·Section 4- Extras
Homework: Homework is assigned on most days. It is essential to your success that you practice the skills you learn in class. Homework is due at the beginning of class the day after it is assigned. Homework may either be collected and graded, or assigned an effort grade.
Notebook: There will be occasional notebook quizzes to check the accuracy of class notes. These quizzes will not be announced in advance and the number of points will vary.
Tests and Quizzes: You will be notified in advance of chapter Tests and Quizzes and are expected to spend time outside of class preparing for them.
Class Rules:
·Bring all materials each day.
· Leaving class for any reason (restroom/locker/etc.) will result in a tardy.
·No eating or drinking in class (except water).
Absence:If you are absent it is your responsibility to get class notes from a classmate and complete any missed assignments. Mrs. Vargo'smake up policy is the same as the school policy (read pg. 6-7 in the student handbook!)
·If you miss class and need help with any new material taught while you were absent, you may make arrangements to come in before or after school for help. Lessons will not be re-taught during class.
·You have two days for every day absent to make up homework assignments.
·If you are only absent on the day of a test or quiz, you must make up the test the day you return.
· If you are absent the day before a test or quiz and no new material was taught, you must take the test as scheduled with the rest of the class.
·Grades will be lowered one letter grade for each late make-up day.
·Make up tests and quizzes may not be taken during class. You must make arrangements to do make-ups before or after school or during study hall. |
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Mathematics—Secondary Education, BAEd
What Is the Study of Mathematics—Secondary?
"Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics-From Everybody Counts: A Report to the Nation on the Future of Mathematics Education (c) 1989 National Academy of SciencesTeaching Mathematics is a challenge, a responsibility, and an opportunity. Learning to teach Mathematics occurs through a variety of means: the study of a wide variety of Mathematics, pedagogical preparation within a mathematical context, formal clinical preparation in education, an extended internship, and continual experiences as a student, learner, and problem solver in Mathematics.
Everyone aspiring to be a Mathematics teacher is aware of the demand for qualified teachers at the secondary level, but there is an even greater need for quality Mathematics teachers—teachers who care about both students and Mathematics, teachers who have a broad and deep understanding of Mathematics, and teachers who are thoroughly professional. The responsibilities are great, but the rewards are even greater.
As a prospective teacher you need to focus on expanding your personal understanding of Mathematics and capitalizing on opportunities to work with pre-college students as a tutor, as a classroom assistant, as a practicum student, and as a novice teacher in your internship.
Are you up for it?
How to Declare:
Students who intend to complete a major in Mathematics are urged to declare the major formally at an early point in their Western career so that a program of study can be planned carefully in collaboration with a departmental advisor |
Mathematics
The mission of the Mathematics Department is to develop and implement curriculum that is scientifically research based and aligned with national, state, and local standards in order to improve student achievement. The department believes that a comprehensive, high quality, mathematics education enables students to increase their mathematical literacy in understanding mathematical principals and further enhances mathematical fluency that is essential for success in the 21st century. |
Ratios, Fractions and Percentage Explained students competing in entrance exams like CAT, MAT & Bank PO
Course instructor has been teaching Mathematics for last 8 years
About the Course
Language of Instruction: English
Course Description
Ace Bank PO, CAT, MAT by building strong basics in Mathematics!
You can not clear entrance examinations like Bank PO, CAT, MAT in Mathematics without building strong basics. A fair percentage of questions in these exams cover topics like ratios, fractions, and percentages.
If you do not have a strong mathematics background or if you find yourself rusty in logical Math, this course will help you learn ratios, fractions and percentages. From basic concepts to solving word problems, this course covers everything. In 3 LIVE interactive online classes, you will clear your basics in ratios, fractions, and percentages. Also, you can attempt 3 online tests to evaluate your performance.
Course instructor Yasmita D Purohit is a post graduate in Applied Mathematics who has taught Mathematics to students for last 8 years. She has structured this course to speed up your calculations in three important topics of Mathematics. Also, you get to practice Mathematics through worksheets and assignments.
Course outline:
Class No.
Topic
Class duration
1.
Fractions
45 minutes
2.
Ratios and fractions
45 minutes
3.
Percentage
45 minutes
About the Instructor
Yasmita Purohit Pondicherry, India
Yasmita D Purohit has taught Mathematics to students for past 8 years. She has done MSc. in Applied Mathematics from Pondicherry University in 1999. She is presently working as an international teacher in Mathematics and Computer in Universal School in Jakarta, Indonesia.
Testimonial
good class. Learnt a new technique..
Came to know differnt approach the same problem
when it is solved on board. |
CliffsQuickReview course guides cover the essentials of your toughest classes. You're sure to get a firm grip on core concepts and key material and be ready for the test with this guide at your side.
Whether you're new to functions, analytic geometry, and matrices or just brushing up on those topics, CliffsQuickReview Precalculus can help. This guide introduces each topic, defines key terms, and walks you through each sample problem step-by-step. In no time, you'll be ready to tackle other concepts in this book such as
Arithmetic and algebraic skills
Functions and their graphs
Polynomials, including binomial expansion
Right and oblique angle trigonometry
Equations and graphs of conic sections
Matrices and their application to systems of equations
Get all you need to know with Super Reviews! Each Super Review is packed with in-depth, student-friendly topic reviews that fully explain everything about the subject. The Pre-Calculus Super Review includes sets, numbers, operations and properties, coordinate geometry, fundamental algebraic topics, solving equations and inequalities, functions, trigonometry, exponents and logarithms, conic sections, matrices, and determinants. Take the Super Review quizzes to see how much you've learned - and where you need more study. Makes an excellent study aid and textbook companion. Great for self-study!Well, the good news is that you can master calculus. It's not nearly as tough as its mystique would lead you to think. Much of calculus is really just very advanced algebra, geometry, and trig. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.
Calculus For Dummies is intended for three groups of readers:
Students taking their first calculus course – If you're enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.
Students who need to brush up on their calculus to prepare for other studies – If you've had elementary calculus, but it's been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies will give you a thorough, no-nonsense refresher course.
Adults of all ages who'd like a good introduction to the subject – Non-student readers will find the book's exposition clear and accessible. Calculus For Dummies takes calculus out of the ivory tower and brings it down to earth.
This useful guide helps both new students and those who need a refresher course to acquire practical skills in calculus through a series of 20 lesson plans that require a minimal time commitment. All key calculus topics are covered, from common functions and their graphs to differentiation, integration, and infinite series. The book contains hundreds of practice exercises without a lot of unnecessary theory or math jargon. Bonus sections offer additional resources and tips for taking standardized tests.
Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or, worse yet, not know where to begin? Have no fear! This hands-on guide focuses on helping you solve the many types of calculus problems you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with limits, continuity, curve-sketching, natural logarithms, derivatives, integrals, infinite series, and more!
100s of Problems!
Step-by-step answer sets clearly identify where you went wrong (or right) with a problem
The inside scoop on calculus shortcuts and strategies
Know where to begin and how to solve the most common problems
Use calculus in practical applications with confidence
Whether you're brushing up on pre-Algebra concepts or on your way toward mastering algebraic fractions, factoring, and functions, CliffsQuickReview Algebra I can help. This guide introduces each topic, defines key terms, and carefully walks you through each sample problem step-by-step. In no time, you'll be ready to tackle other concepts in this book such as
Whether you're new to limits, derivatives, and integrals or just brushing up on your knowledge of the subject, CliffsQuickReview Calculus can help. This guide covers calculus topics such as limits at infinity, differential rules, and integration by parts. You'll also tackle other concepts, including
CliffsQuickReview Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry — whether you need a supplement to your textbook and classes or an at-a-glance reference. Trigonometry isn't just measuring angles; it has many applications in the real world, such as in navigation, surveying, construction, and many other branches of science, including mathematics and physics. As you work your way through this review, you'll be ready to tackle such concepts as
One easy step of a time, this book will teach you the key statistical techniques you'll need for finance, quality, marketing, the social sciences, or just about any other field. Each technique is introduced with a simple, jorgon-free explanation, practical examples, and hands-on guidance for solving real problems with Excel or a TI-83/84 series calculator, including Plus models. Hate math? No sweat. You'll be amazed how little you need! For those who do have an interest in mathematics, optional "Equation Blackhoard" sections review the equations that provide the foundations for important concepts. |
2.1 Functions and Graphs
Introduction Using the objects and the persons around us, it is easy to make up a rule of correspondence that associates, or pairs, the members, or elements, of one set with the members of another set. For example, to each social security number there is a person, to each car registered in the state of California there is a license plate number, to each book there corresponds at least one author, to each state there is a governor, and so on. A natural correspondence occurs between a set of 20 students and a set of, say, 25 desks in a classroom when each student selects and sits in a different desk. In mathematics we are interested in a special type of correspondence, a single-valued correspondence, called a function. |
Because of the highly visual nature of the subject matter, this course is available exclusively on video. It features hundreds of visual elements, including step-by-step walkthroughs of mental math problems, as well as graphics and illustrations. It also features more than 350 visual elements to help you optimize your brain fitness and enhance your learning experience, including detailed diagrams and x-rays of the brain at work, vivid images and graphics, and on-screen text.
Optimizing Brain Fitness & Secrets of Mental Math (Set)
COURSE DESCRIPTION
Tap into your hidden mental potential with this engaging two-course set filled with insights and exercises designed to strengthen and enhance the power of your brain. First, discover the secrets to increasing your brain's ability to meet everyday challenges with Optimizing Brain Fitness. Then, gain all the essential skills, tips, and tricks you need to solve a range of mathematical problems right inside your head with The Secrets of Mental Math.
Course
1
of
2:
Secrets of Mental Math Professor
Professor Arthur T. Benjamin,
Harvey Mudd College Ph.D., Johns Hopkins University
Improve and expand your math potential—whether you're a corporate executive or a high-school student—in the company of Professor Arthur T. Benjamin, one of the most entertaining members of The Great Courses faculty. The Secrets of Mental Math, his exciting 12-lecture course, guides you through all the essential skills, tips, and tricks for enhancing your ability to solve a range of mathematical problems right in your head. Along the way, you'll discover how mental mathematics is the gateway to success in understanding and mastering higher fields, including algebra and statistics.
Course Lecture Titles
12
Lectures
30
minutes/lecture
1.
Math in Your Head!
Dive right into the joys of mental math. First, learn the fundamental strategies of mental arithmetic (including the value of adding from left to right, unlike what you do on paper). Then, discover how a variety of shortcuts hold the keys to rapidly solving basic multiplication problems and finding squares.
7.
Intermediate Multiplication
Take mental multiplication to an even higher level. Professor Benjamin shows you five methods for accurately multiplying any two-digit numbers. Among these: the squaring method (when both numbers are equal), the "close together" method (when both numbers are near each other), and the subtraction method (when one number ends in 6, 7, 8, or 9).
2.
Mental Addition and Subtraction
Professor Benjamin demonstrates how easily you can mentally add and subtract one-, two-, and three-digit numbers. He also shows you shortcuts using the complement of a number (its distance from 100 or 1000) and demonstrates the uses of mental addition and subtraction for quickly counting calories and making change.
8.
The Speed of Vedic Division
Vedic mathematics, which has been around for centuries, is extremely helpful for solving division problems—much more efficiently than the methods you learned in school. Learn how Vedic division works for dividing numbers of any length by any two-digit numbers.
3.
Go Forth and Multiply
Delve into the secrets of easy mental multiplication—Professor Benjamin's favorite mathematical operation. Once you've mastered how to quickly multiply any two-digit or three-digit number by a one-digit number, you've mastered the most fundamental operations of mental multiplication and added a vital tool to your mental math tool kit.
9.
Memorizing Numbers
Think that memorizing long numbers sounds impossible? Think again. Investigate a fun—and effective—way to memorize numbers using a phonetic code in which every digit is given a consonant sound. Then practice your knowledge by trying to memorize the first 24 digits of pi, all of your credit card numbers, and more.
4.
Divide and Conquer
Turn now to the last fundamental operation of arithmetic: division. Explore a variety of shortcuts for dividing by one- and two-digit numbers; learn how to convert fractions such as 1/7 and 3/16 into decimals; and discover methods for determining when a large number is divisible by numbers such as 3, 7, and 11.
10.
Calendar Calculating
The fun continues in this lecture with determining the day of the week of any date in the past or in the future. What day of the week was July 4, 2000? How about February 12, 1809? You'd be surprised at how easy it is for you to grasp the simple mathematics behind this handy skill.
5.
The Art of Guesstimation
In most real-world situations—such as figuring out sales tax or how much to tip—you don't need an exact answer but just a reasonable approximation. Here, develop skills for effectively estimating addition, subtraction, multiplication, division, and square roots.
11.
Advanced Multiplication
Professor Benjamin shows you how to do enormous multiplication problems in your head, such as squaring three-digit and four-digit numbers; cubing two-digit numbers, and multiplying two-digit and three-digit numbers. While you may not frequently encounter these large problems, knowing how to mentally solve them cements your knowledge of basic mental math skills.
6.
Mental Math and Paper
Sometimes we encounter math problems on paper in our daily lives. Even so, there are some rarely taught techniques to help speed up your calculations and check your answers when you are adding tall columns of numbers, multiplying numbers of any length, and more.
12.
Masters of Mental Math
Professor Benjamin concludes his exciting course by showing how you can use different methods to solve the same problem; how you can find the cube root of large perfect cubes; how you can use the techniques you've learned and mastered in the last 11 lectures; and more.
Course Lecture Titles
12
Lectures
30
minutes/lecture
1.
How Your Brain Works
In order to best optimize your brain fitness, it's important to understand how the brain's circuitry works. After a brief introduction to the course, Professor Restak guides you through a range of intriguing topics, including the principles of brain operation, the organization of the brain, patterns of brain growth, and more.
7.
Exercising Your Working Memory
Focus now on working memory—the most important memory process of all and one that involves manipulating stored information. After an overview of the topic, dive into a series of engaging exercises that use your creativity, your powers of observation, and your heightened awareness to enhance and improve your working memory.
2.
How Your Brain Changes
Your brain and your intelligence can change throughout your life span. Here, look closer at the way changes in your brain can improve the way you function in your day-to-day life. Also, explore how a series of visual, sensory, and spatial exercises demonstrate the powerful effects of brain plasticity.
8.
Putting Your Senses to Work
Imaginative memory techniques—such as mnemonic devices and personal associations—have been used to improve memory for over 1,000 years. Try your hand at some of them right here, including "chunking" numbers to aid in number recall, creating a vivid story to memorize words, drawing free-form designs, and playing mental chess.
3.
Care and Feeding of the Brain
You can optimize your brain function by paying attention to three key habits: what you eat, how well you sleep, and how much you exercise. Ponder the science behind this three-pronged approach to caring for your brain, and come away with helpful tips you can apply to your own lifestyle.
9.
Enlisting Your Emotional Memory
Turn now to an aspect of memory we don't usually consider when thinking about the subject: emotional memory. How did scientists uncover this specific aspect of memory? How does it actually work? And what kinds of playful exercises can you perform to help you relive the emotional experience of your past?
4.
Creativity and the Playful Brain
What's the connection between daydreaming and creativity? What are four steps for increasing your creativity? Which puzzles are the best for optimizing your brain function—and how can you more efficiently solve them? Learn the answers to these and other questions in this fascinating lecture on creativity and the brain.
10.
Practicing for Peak Performance
Exceptional performers aren't born with "superior brains." Rather, anyone—thanks to brain plasticity—can achieve high performance levels in an area of interest through deliberate practice. Focus here on two aspects of deliberate practice: remaining fully aware of what you're doing, and concentrating on the most difficult aspects of your performance.
11.
Taking Advantage of Technology
Take a closer look at the impact of modern technology on how our brains function. You'll explore the positive and negative effects of electronic journals, personal computers, and more—with a lengthy discussion on the impact of one of today's most powerful and controversial influences on brain function: video games.
6.
Enhancing Your Memory
In the first of three lectures devoted to memory, Dr. Restak proves just how essential memory is to your brain's optimal functioning. After surveying the details of memory and its roots in the hippocampus, learn ways to sharpen your sense memory and augment both your short-term and long-term general memory.
12.
Building Your Cognitive Reserve
Professor Restak concludes his course with ways to immediately start optimizing your brain fitness. These include trying new and unexpected things, learning in an informal and self-directed manner, keeping things in perspective, opting to prioritize instead of multitask, developing an appreciation for art and music, and—surprisingly—preparing home-cooked meals. |
1 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: Algebra I Standards
ALGEBRA I CONTENT STANDARDS 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 properties when applicable.
Modular Arithmetic Sample draft: Do not cite or quote Page 12 2 Modular Arithmetic 2.1Introduction Studying remainders and their properties under the operations From a teaching perspective, "remainder arithmetic"offers students opportunities to study the structure of number systems on a small ... |
PLEASE NOTE: THE SPECIAL ORDER TAKES 4-5 BUSINESS DAYS TO SHIP OUT. free upgrade to 1st class shipping with tracking. (8-14 days to arrive.)
1. ID # 6007
2. $3.99 FOR MEDIA MAIL SHIPPING
3. the listing contains no books. it's only a TEACHER'S CD ROM. contents on it:
=====HARDCOVER TEACHER'S EDITION. FULL ANSWERS FOR EVERY PROBLEM IN THE STUDENT TEXTBOOK, EVEN AND ODD.
=====whole set teaching resources, workbook answer key, quizzes tests with answers.
4. THE ITEM IS USED TO THE FOLLOWING STUDENT TEXTBOOKS:
===0030700523 (2004)
===003066053X (2003)
===0030522196 (2001)
THE STUDENT TEXTBOOK IS NOT INCLUDED |
arson's market-leading text, PRECALCULUS is known for delivering sound, consistently structured explanations and exercises of mathematical concepts to expertly prepare students for the study of calculus. With the ninth edition, the author continues to revolutionize the way students learn material by incorporating more real-world applications, ongoing review, and innovative technology. How Do You See It? exercises give students practice applying the concepts, and new Summarize features, Checkpoint problems, and a Companion Website reinforce understanding of the skill sets to help students better prepare for tests. |
In grades 9-12, the
mathematics curriculum should include the informal exploration of
calculus concepts from both a graphical and a numerical perspective
so that all students can--
determine maximum
and minimum points of a graph and interpret the results in problem
situations;
investigate limiting
processes by examining infinite sequences and series and areas
under curves;
and so that, in addition,
college-intending students can--
understand the
conceptual foundations of limit, the area under a curve, the rate
of change, and the slope of a tangent line, and their applications
in other disciplines;
analyze the graphs
of polynomial, rational, radical, and transcendental functions.
Focus
This standard does not
advocate the formal study of calculus in high school for
all students or even for college-intending students. Rather, it
calls for opportunities for students to systematically, but informally,
investigate the central ideas of calculus--limit, the area under
a curve, the rate of change, and the slope of a tangent line--that
contribute to a deepening of their understanding of function and
its utility in representing and answering questions about real-world
phenomena.
Most of the mathematics
described in the other 9-12 standards involve finite processes,
such as determining a sequence of transformations that maps a figure
onto a congruent figure or approximating a zero of a polynominal
function using an iterative technique. In contrast, the concept
of limit and its connection with the other mathematical topics in
this standard is based on infinite processes. Thus, explorations
of the topics proposed here not only extend students' knowledge
of function characteristics but also introduce them to another mode
of mathematical thinking.
Instruction should be highly
exploratory and based on numerical and geometric experiences that
capitalize on both calculator and computer technology. Instructional
activities should be aimed at providing students with firm conceptual
underpinnings of calculus rather than at developing manipulative
techniques.
Discussion
The development of the
calculus represents one of the great intellectual accomplishments
in human history; perhaps the greatest achievement in the application
of mathematics is the use of calculus in physics during the first
third of this century. Today, methods of calculus are applied increasingly
in the social and biological sciences, and in business as well.
As students explore the topics proposed in this standard, it is
important that they develop an awareness of, and appreciation for,
the historical origins and the cultural contributions of the calculus.
The topics proposed for
investigation in this standard should be developed as natural extensions
of ideas students have previously encountered. The study of finite
sequences and series recommended in the discrete-mathematics standard
leads naturally to consideration of the corresponding infinite cases
and concepts associated with limiting processes. Considerations
of infinite sequences and series can occur at many different levels
of abstraction and formalism. Consider, for example, the series
At a very concrete level,
this series can be summed in parts, as shown in figure
13.1.
Fig. 13.1.An informal approach to infinite series
At a somewhat higher but
still intuitive level, the series can be summed by investigating
the limit of the sequence of partial sums (either by using a calculator
or a simple looping computer program or by applying the formula
for the sum of a finite geometric series). An understanding of the
concept of limit in contexts such as this should in turn contribute
to the meaningful development of the remaining topics in this standard.
Similarly, it is important
that all students recognize how the concept of the area under a
curve builds on and extends their previous experiences with areas
of geometric figures. College-intending students also should recognize
how the rate of change builds on and extends their experiences with
uniform motion and associated rates in algebra and trigonometry,
and how the slope of a tangent to a curve generalizes the notion
of the slope of a line as developed in algebra.
Many of these concepts
can be approached through informal activities that focus on the
understanding of interrelationships. For example, the concepts of
slope, derivative, velocity, and acceleration could be addressed
through experiences such as the following:
Given the velocity-time
graph shown in figure 13.2, construct reasonable
distance-time and acceleration-time graphs for the same system.
Note that correct responses
may vary; for example, the distance-time graph could be translated
vertically.
Computing technology makes
the fundamental concepts and applications of calculus accessible
to all students. The area under a finite portion of a curve, for
example, can be approximated geometrically by partitioning the region
into rectangles with bases of equal length and heights given by
function values at an endpoint of each base. Using a calculator
or a computer-based algorithm, students can easily sum the areas
to obtain a numerical approximation of the desired area. This approximation
can then be sharpened by using sequences of rectangles whose bases
are made to decrease toward zero. Project work might require students
to investigate other ways of partitioning the region so as to obtain
a more precise estimate. (A description of a process using trapezoids
can be found in the standard on mathematical
connections.)
All students could use
a graphing utility to investigate and solve optimization problems,
including the maximum-minimum problems traditionally associated
with the first college-level course in calculus, without computing
a derivative. (For an example, see
Examples of Content Differentiation,) A great deal of mathematical
understanding is reinforced in the context of solving these problems:
data analysis, problem formulation, mathematical modeling, geometric
topics, translation across multiple representations, and validation.
Using interactive graphing
utilities, college-intending students could examine other characteristics
of the graphs of functions, including continuity, asymptotes, end
behavior (i.e., behavior as |x| ), and concavity. Moreover,
such analysis can be applied with equal ease to the graphs of a
variety of complicated functions and to curves specified by polar
and parametric equations.
Computing technology also
permits the foreshadowing of analytic ideas for college-intending
students. From a computer-graphics perspective, for example, a differentiable
function can be viewed as a function having the property that a
small portion of its graph, when highly magnified, approximates
a line segment. The "zoom in" feature permits students
to magnify the graph of a function over a small interval to view
the approximate segment representation and to compute the gradient
(slope). Using the computer to plot successive gradients for a series
of short intervals (see fig. 13.4) suggests
a process that derives functions from functions.
Fig. 13.4.Using computer graphics to see that the derivative of y
= sin x is y = cos x
Although developing a foundation
for the future study of calculus remains a goal of the 9-12 curriculum
for college-intending students, equally important is the development
of prerequisite understandings for further study of statistics,
probability, and discrete mathematics. This standard calls for a
new balance of skills, concepts, and applications in that portion
of the curriculum traditionally associated with preparation for
calculus. Instead of devoting large blocks of time to developing
a mastery of paper-and-pencil manipulative skills, more time and
effort should be spent on developing a conceptual understanding
of key ideas and their applications. All students should have the
benefit of a computer-enhanced introduction to some of the types
of problems for which calculus was developed. |
Courses & Descriptions
*(ISAT test scores, teacher recommendations and transcript grades are used to place students in math courses.)*
2105 - PRE-ALGEBRA (9) 2 Credits/2 Semesters
This course is offered for students that have NOT yet met the criteria for entering Algebra 1. Topics include simple equations and inequalities, exponent rules, linear graphing, radicals, and algebraic evaluations. The course counts towards elective credit and does not meet the Math requirements for high school graduation.
2115 - ISAT MATH (11,12) 2 Credits/2 Semesters
This course is for juniors who have not passed the ISAT math test.
2120 - ALGEBRA AB (9,10) 2 Credits/2 Semesters
This course covers the same concepts as the Algebra 1 course, with the only difference being the pace. It takes 2 semesters to complete this algebra series. Topics covered are integers, properties, order of operations, solving equations and inequalities, word problems, proportions, percents, trigonometry, plotting points, relations, functions, graphing and writing line equations.
The student will examine the complete real number system and its structure through the development of algebraic language and skills. An introduction to logic, exposure to algebraic proofs and opportunities to solve realistic problems are important ingredients of this course. Major skills covered are solving linear and quadratic equations.
2210 - APPLIED MATH 1 (10,11,12) 2 Credits/2 Semesters
Students will be given a lab class, "hands on" opportunity, to focus on arithmetic operations, problem-solving techniques, estimation of answers, measurement skills, algebra, geometry, data handling, statistics, and computers to solve problems. Prerequisite: Passed Pre-Algebra (with a C or better) or completed Algebra.
2220 - APPLIED MATH 2 (11,12) 2 Credits/2 Semesters
Students will be given a lab class, "hands on" opportunity to focus on scientific notation, vectors, precision, formulas used to solve real-life problems, use nonlinear and linear equations to solve problems, and work with probabilities and statistics. Prerequisite: Passed Applied math 1 or completed Geometry.
2230 - APPLIED MATH 3 (11,12) 2 Credits/2 Semesters
This is a continuation of the Applied Math course with emphasis on trigonometry and analytical geometry. It involves labs and activities as well as problems taken from the work environment. Prerequisite: Algebra 2 or Applied Math 2.
2320 - GEOMETRY (9,10,11,12) 2 Credits/2 Semesters
Geometry is a Math course involing measurement and dealing with shapes. Some of the concepts studied in elementary Algebra will be helpful to the student's understanding of Geometry. Geometry is valuable because of its wide variety of applications to other subjects such as astronomy, art and chemistry. Through the use of logic and imagination, the student will examine the postulated structure of Euclidean Geometry. Prerequisite: Passing Algebra with a C or better.
2330 - ALGEBRA 2 (9,10,11,12) 2 Credits/2 Semesters
Advanced Algebra provides the finishing touch that will unlock the mysteries and magic of science, engineering, architecture, design, health science, and of course, mathematics. Some concepts presented in Geometry will be most helpful to the student of Advanced Algebra. Major Skills covered are a study of higher degree polynomials, logarithms and exponents, and conics. Prerequisite: Passing Geometry with a C or better.
This course enhancees the Algebra 2 curriculum with a focus on application problems. The class combines some topics that are not covered in a traditional Algebra 2 class with topics that they have already learned to create a deeper understanding of Algebra 2 and its application in the world. Prerequisite: Successful completion of Algebra 2.
This course wil cover functions, matrices, algebra and trignometry. It will also cover statistics and probability. Students will learn how to use a graphing calculator to enhance the understanding of the concepts. Pre-Calculus will prepare students who are planning on attending a two or four year university. This course will also prepare students who are planning on taking IB Mathematics SL. Prerequisite: Completed Algebra II with a "C" or better
2421 - AP CALCULUS (12) 2 Credits/2 Semesters
The objective of AP Calculus is to give students an appreciation of Calculus and to build a strong foundation that will allow them to succeed in futuree mathematics courses. Advanced Placement Calculus -AB satisfies the requirements designed by the College Board and is equilvalent to one semester of college Calculus. There will be an Advanced Placement Calculus-AB Exam in May in which all students are expected to participate in. The major topics of this course are the differentiation and integration of algebraic and transcendental functions with applications. Should have successfully completed Pre Calculus with a B or higher in order to be in AP Calculus-AB. Students must be familiar with the properties of functions, the algebra of functions, and the graphs of functions.
2449 - IB MATH STUDIES, SL - YR (11,12) 2 Credits/2 Semesters
This course will cover linear/non-linear functions, probability and statistics, sequences and series, some geometry and trignometry, as well as an introduction to differential calculus. The purpose of Math Studies SL is to provide students at Lake City High School, who are pursuing non-Math related careers, an opportunity for mathematical growth and discovery. A project will be added requirement of the course and will stretch over the course of the whole year. Prerequisite: Completed Pre-Calculus with a "C" or better.
2441 - IB MATHEMATICS SL (11,12) 2 Credits/2 Semesters
The first year topics that will be covered are functions, matrices, algebra, statistics and probability, circular functions and trigonometry. The second year topics that will be covered are vectors, advanced statistics and probability, and calculus. Emphasis will be on both understanding and application. Students will learn the international symbols, the use of the graphing calculator, and mathematical writing skills. During the second year the history of mathematics will be used to connect the topics. Also, students will be required to submit two projects. Summer homework will be assigned for both classes. Prerequisits for 2410 - HON PRE-CALC (10,11): Completed Algebra II with a "C" or better. Prerequisits for 2441 - IB MATHEMATICS SL (11,12): Completed Mathematics 1 with a "C" or better.
2510 STATISTICS Sem (10,11,12) 1 Credit/1 Semester
This course will introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes:
This course draws connections between all aspects of the statistical process, including design, analysis, and conclusions. Additionally, using the vocabulary of statistics this course will teach students how to communicate statistical methods, results and interpretations. Students will learn how to use graphing calculators and read computer output in an effort to enhance the development of statistical understanding. Prerequisite:Successful completion of Algebra 2.
7755 - PERSONAL FINANCIAL PLANNING (ME0100) (12) 1 Credit/1 Semester
The students will learn the financial planning process including goals, credit, investing, retirement, insurance, safety net, taxes and creating a personal financial portfolio.
2660 - NIC MATH 025 (11,12) 3 NIC Credits/1 LCHS Credit/1 Semester Introduction to mathematical concepts dealing with signed numbers, variable, polynomials, exponents, factoring, solving and graphing first-degree equations, and inqualities. The course also introduces solving factorable second-degree equations. It emphasizes the practical applications of these concepts. The course provides important skill-building for those who have not taken or have had difficulty with high school algebra. Prerequisite: COMPASS Placement Test required.
2665 - NIC MATH 108 (11,12) 4 NIC Credits/ 1 LCHS Credit/1 Semester Development of mathematical concepts beyond MATH 025 or first year high school algebra. It includes linear and quadratic equations, algebraic fractions, radicals, circles and parabolas, complex numbers, functions, and logarithms. This is an emphasis on the application of these skills. The course provides important skill building for entry into collebe-level math courses. Enrollment is based on placement test results. Prerequisite: COMPASS Placement Test required. |
Business Math For Dummies
Synopsis
The essential desk reference for every business professional or student
This easy-to-understand resource explains complex mathematical concepts and formulas and offers clear examples of how they relate to real-world business situations. Featuring practical practice problems to help readers hone their skills, it covers such key topics as working with percents to calculate increases and decreases, using basic algebra to solve proportions, and using basic statistics to analyze raw data. Readers will also find solutions for finance and payroll applications, including reading financial statements, calculating wages and commissions, and strategic salary planning.
Business Math For Dummies
eBook Information
ISBN: 9780470397398 |
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